EME    3flS 


NON-EUCLTDEAN 
GEOMETRY 


BY 


HENRY   PARKER   MANNING,  Pii.D. 

ASSISTANT  PROFESSOR  OF  PURE  MATHEMATICS 
IN  HUoWN  UNIVERSITY 


BOSTON,  U.S.A. 
GINN   &   COMPANY,  PUBLISHERS 

<L  be  .atbrncTui 
1901 


, 


' 


COPYRIGHT,  1901,  BY 
HENRY  PARKER  MANNING 


ALL  BIGHTS  RESERVED 


CONTEXTS 


IN  i  i;c>i>r<-Tlox 


(  1IA1TKR    I 
PANGEOMETKY 

I.      PROPOSITION-     M  1-1  SIMM.     «»M.I      <»s     nn;      I'IMSCM-M:     or 

Si  i-i.iM'oM  i  II.N     .          .          .          .          .          .          .          .          .  '•*> 

II.      I'uoro-i  i  ION-    \\iin  ii    MM.   TIM  i.    i'»i:    1{  i  .«.  i  i;  i<  1  1  c    I-"HMIM-   .  0 

III.     THE  Tiiuti:    ll\  r«  .....  BKf         ...                             .          .  11 

CHAPTEB    II 

rilK    IIVI'KKI'.OLK      (.Ko.MKTKV 


I.       l'\i:  vi  I  I  i     1-iM  -       .........  ^1 

II.       BoUKDART-CuRVEi    UTD  SUBVACM,    \M«   K-.'i  1  1  -i-  i  v  M  -(  'i  i;  \  i  - 

\\1.    Si    1M    V«    I-.S       .........  J;i 

111.     TRIOOKOMEI  IM.  \i    K..KMI  i  i   .  .        .        .        .  5:> 

CHAPTEK    111 

TIII-:  ELLIPTIC  «ii-:oMi-;THy     .       .       .  <;•-> 

CHA1TKII    IV 
ANALYTIC    NON-EUCLIDKAN    (  .KnM  KTH  V 

I.     HYPERBOLIC  ANALYTIC  GEOMETRY          .....  09 

II.     ELLIPTIC  ANALYTIC  GEOMETRY       ......  *<> 

III.      ELLIPTIC  SOLID  ANALYTIC  GEOMETRY    .....  s<> 

Hl-H'IMi    VI.    NoTK        ..........  (.>1 


PREFACE 

NON-EUCLIDEAN  Geometry  is  now  recognized  as  an  impor- 
tant branch  of  Mathematics.  Those  who  teach  Geometry 
should  have  some  knowledge  of  this  subject,  and  all  who 
are  interested  in  Mathematics  will  find  much  to  stimulate 
them  and  much  for  them  to  enjoy  in  the  novel  results  and 
views  that  it  presents. 

This  book  is  an  attempt  to  give  a  simple  and  direct 
account  of  the  Non-Euclidean  Geometry,  and  one  which 
presupposes  but  little  knowledge  of  Mathematics.  The  first 
three  chapters  assume  a  knowledge  of  only  Plane  and  Solid 
Geometry  and  Trigonometry,  and  the  entire  book  can  be  read 
by  one  who  has  taken  the  mathematical  courses  commonly  given 
in  our  colleges. 

No  special  claim  to  originality  can  be  made  for  what  is 
published  here.  The  propositions  have  long  been  estab- 
lished, and  in  various  ways.  Some  of  the  proofs  may  be 
new,  but  others,  as  already  given  by  writers  on  this  subject, 
could  not  be  improved.  These  have  come  to  me  chiefly 
through  the  translations  of  Professor  George  Bruce  Halsted 
of  the  University  of  Texas. 

I  am  particularly  indebted  to  my  friend,  Arnold  B.  Chace, 
Sc.D.,  of  Valley  Falls,  K.  I.,  with  whom  I  have  studied  and 
discussed  the  subject. 

HENRY    P.    MANNING. 

PROVIDED  i  .  .January,  1901. 

iii 

888489 


NON-EUCLIDEAN    GEOMETRY 


INTRODUCTION 

THE  axioms  of  Geometry  were  formerly  regarded  as  laws 
of  thought  which  an  intelligent  mind  could  neither  deny  nor 
investigate.  Not  only  were  the  axioms  to  which  we  have 
been  accustomed  found  to  agree  with  our  experience,  but 
it  was  believed  that  we  could  not  reason  on  the  supposition 
that  any  of  them  are  not  true.  It  has  been  shown,  however, 
that  it  is  possible  to  take  a  set  of  axioms,  wholly  or  in  part 
contradicting  those  of  Euclid,  and  build  up  a  Geometry  as 
consistent  as  his. 

We  shall  give  the  two  most  important  Non-Euclidean 
Geometries.*  In  these  the  axioms  and  definitions  are  taken 
as  in  Euclid,  with  the  exception  of  those  relating  to  parallel 
lines.  Omitting  the  axiom  on  parallels,!  we  are  led  to  three 
hypotheses  ;  one  of  these  establishes  the  Geometry  of  Euclid, 
while  each  of  the  other  two  gives  us  a  series  of  propositions 
both  interesting  and  useful.  Indeed,  as  long  as  we  can  exam- 
ine but  a  limited  portion  of  the  universe,  it  is  not  possible  to 
prove  that  the  system  of  Euclid  is  true,  rather  than  one  of 
the  two  Non-Euclidean  Geometries  which  we  are  about  to 
describe. 

We  shall  adopt  an  arrangement  which  enables  us  to  prove 
first  the  propositions  common  to  the  three  Geometries,  then 
to  produce  a  series  of  propositions  and  the  trigonometrical 
formulae  for  each  of  the  two  Geometries  which  differ  from 

*  See  Historical  Note,  p.  '.»:].  f  See  p.  91. 

1 


2  NON-EUCLIDEAN  GEOMETRY 

,'thkfc  of  Euclid,;  and  by  analytical  methods  to  derive  some  of 
their  most  striking  properties. 

)/IWfc{cfo.:u<>t  propose;  to  investigate  directly  the  foundations 
of  Geometry,  nor  even  to  point  out  all  of  the  assumptions 
which  have  been  made,  consciously  or  unconsciously,  in  this 
study.  Leaving  undisturbed  that  which  these  Geometries 
have  in  common,  we  are  free  to  fix  our  attention  upon  their 
differences.  By  a  concrete  exposition  it  may  be  possible  to 
learn  more  of  the  nature  of  Geometry  than  from  abstract 
theory  alone. 

Thus  we  shall  employ  most  of  the  terms  of  Geometry  with- 
out repeating  the  definitions  given  in  our  text-books,  and 
assumt  that  the  figures  defined  by  these  terms  exist.  In 
particular  we  assume  : 

I.  The  existence  of  straight  lines  determined  by  any  two 
points,  and  that  the  shortest  path  between  two  points  is  a 
straight  line. 

II.  The  existence  of  planes  determined  by  any  three  points 
not  in  a  straight  line,  and  that  a  straight  line  joining  any  two 
points  of  a  plane  lies  wholly  in  the  plane. 

III.  That  geometrical  figures  can   he.   mored  <iho«t  iritJtottt 
changing  their  shape  or  size. 

IV.  That  a  point  moving  along  a  line  from   one  position  to 
another  passes  through  every  point  of  the  line  between,  and 
that  a  geometrical  magnitude,  for  example,  an  angle,  or  the 
length  of  a  portion  of  a  line,  varying  from  one  value  to  another, 
passes  through  all  intermediate  values. 

In  some  of  the  propositions  the  proof  will  be  omitted  or 
only  the  method  of  proof  suggested,  where  the  details  can  be 
supplied  from  our  common  text- books. 


CHAPTER   I 
PANGEOMETRY 

I.    PROPOSITIONS   DEPENDING   ONLY   ON   THE   PRINCIPLE 
OF  SUPERPOSITION 

1.'  Theorem.    If  one  straight  line  meets  another,  the  sum  of 
r/t,    nljacent  angles  formed  is  equal  to  two  right  angles. 

2.  Theorem.    If  two   straight    lines   intersect,   the   vertical 
titi<//t'8  are  equal. 

3.  Theory—     Two  triangles  are  equal  if  they  have  a  */</> 
<iml  two  adjacent  <//////*•*.  <>/'  tiro  */,/,  .v  <nt</  tin'  fn,-/n<lrtl  <//////«•, 

»/  »ne  equal,  respectively,  to  the  corresponding  parts  of  (//<• 
other. 

4.  Theorem.    In  an  isosceles  triati'/lc  t/n-  ///////*'*  opposite  th<- 
r<///<tl  sides  are  equal. 

Bisect  the  angle  at  the  vertex  and  use  (3). 

5.  Theorem.    The  perpendiculars    erected    at    the    middle 
fi'iiids  of  the  sides  of  a  triangle  ///<  >•/  ///  a  point  if  tir<>  »/ 
them  meet,  <tn</  tin*  [>oint  is  the  centre  of  a  circle  f/xtf  <-,m 

i-nHijh  tin-  (///-»  /-,  rfi'-.'s  of  the. triangle. 
c 


Proof.    Suppose  Eu  and  FO  meet  at  O.     The  triangles  A  TO 
and   /;/•'/>  are  equal  by  (3).     Also,  A  En  and   <'EO  are  equal. 

3 


4  NON-EUCLIDEAN   GEOMETRY 

Hence,  CO  and  BO  are  equal,  being  each  equal  to  AO.  Th. 
triangle  B CO  is,  therefore,  isosceles,  and  OD  if  drawn  bisect- 
ing the  angle  BOC  will  be  perpendicular  to  BC  at  its  middle 
point. 

6,  Theorem.    In  a  circle  the  radius  bisecting  an  angle  at 
the  centre  is  perpendicular  to  the  chord  which  subtends  the 
angle  and  bisects  this  chord. 

7,  Theorem.    Angles  at  the  centre  of  a  circle  are  propor- 
tional to  the  intercepted  arcs  and  may  be  measured  by  them. 

8,  Theorem.    From  any  point  without  a  line  a  perpendicu- 
lar to  the  line  can  be  drawn.  P 

Proof.    Let  P'  be  the  position  which  P  would 
take  if  the  plane  were  revolved  about  AB  into  A— 
coincidence  with  itself.     The  straight  line  PP1 
is  then  perpendicular  to  AB.  p, 

9,  Theorem.    If  oblique  lines  drawn  from  a  point  in  a  per- 
pendicular to  a  line  cut  off  equal  distances  from  the  foot  of 
the  perpendicular,  they  are  equal  and  make  equal  angles 
with  the  line  and  with  the  perpendicular. 

10,  Theorem.    If  two  lines  cut  a  third  at  the  same  angle, 

'F 


L 


/( 


'E 

that  is,  so  that  corresponding  angles  are  equal,  a  line  can  be 
drawn  that  is  perpendicular  to  both. 


PROPOSITIONS  PROVED  BY   SUPERPOSITION 


Proof.  Let  the  angles  FMB  and  MND  be  equal,  and  through 
H,  the  middle  point  of  MN,  draw  LK  perpendicular  to  CD ; 
then  LK  will  also  be  perpendicular  to  AB.  For  the  two 
triangles  LMH  and  KNH  are  equal  by  (3). 

11.  Theorem.  If  two  equal  lines  in  a  plane  are  erected  per- 
pendicular to  a  given  line,  the  line  joining  their  extremities 
makes  equal  angles  with  them  and  is  bisected  at  right  angles 
by  a  third  perpendicular  erected  midway  between  them. 


I) 


^UC 

(Let 


II 


B 


Let  AC  and  BD  be  perpendicular  to  AB,  and  suppose  AC 
and  BD  equal.  The  angles  at  C  and  D  made  with  a  line  join- 
ing these  two  points  are  equal,  and  the  perpendicular  HK 
erected  at  the  middle  point  of  AB  is  perpendicular  to  CD  at 
its  middle  point. 

Proved  by  superposition. 

12.  Theorem.  Criven  as  in  the  last  proposition  tivo  perpen- 
diculars and  a  third  perpendicular  erected  midway  between 
them;  any  line  cutting  this  third  perpendicular  at  right 
angles,  if  it  cuts  the  first  two  at  all,  will  cut  off  equal 
lengths  on  them  and  make  equal  angles  with  them. 

Proved  by  superposition. 

Corollary.  The  last  two  propositions  hold  true  if  the  angles 
at  A  and  B  are  equal  acute  or  equal  obtuse  angles,  HK  being 


6  NON-EUCLIDEAN  GEOMETRY 

perpendicular  to  AB  at  its  middle  point.  If  AC  =  BD,  the 
angles  at  C  and  D  are  equal,  and  HK  is  perpendicular  to 
CD  at  its  middle  point  •  or,  if  CD  is  perpendicular  to  HK 

C  K  D 


A          II         B 

at  any  point,  K,  and  intersects  A  C  and  BD,  it  will  cut  off  equal 
distances  on  these  two  lines  and  make  equal  angles  ivith  them. 

II.    PROPOSITIONS   WHICH   ARE   TRUE   FOR    RESTRICTED 
FIGURES 

The  following  propositions  are  true  at  least  for  figures 
whose  lines  do  not  exceed  a  certain  length.  That  is,  if  there 
is  any  exception,  it  is  in  a  case  where  we  cannot  apply  the 
theorem  or  some  step  of  the  proof  on  account  of  the  length  of 
some  of  the  lines.  For  convenience  we  shall  use  the  word 
restricted  in  this  sense  and  say  that  a  theorem  is  true  for 
restricted  figures  or  in  any  restricted  portion  of  the  plane. 


1.  Theorem.    The  exterior  anglv  of  a   triangle  is  greater 
than  either  opposite  interior  angle  (Euclid,  I,  16). 


PROPOSITIONS  TRUE  FOR  RESTRICTED  FIGURES        7 

Proof.  Draw  AD  from  .4  to  the  middle  point  of  the  oppo- 
site side  and  produce  it  to  E,  making  DE  =  AD.  The  two 
triangles  A  DC  and  EBD  are  equal,  and  the  angle  7*727),  being 
greater  than  the  angle  EBD.  is  greater  than  f  '. 

Corollary.     At  li'uxt  ttro  out/It's  <>f  <i  trio  n<il>'  n  re 


2.  Theorem.     //'  ///•»  '//>///''*  "/  <i  //•/<//<///*•  tin-  »•///////.  ///>- 
an  "{ii'il  <ind  th>'  trian<jh  is  isosc*-/*  *. 


c 


Proof.  The  perpendicular  erected  at  the  middle  point  of  the 
base  divides  the  triangle  into  two  figures  which  may  be  made 
to  coincide  and  are  equal.  This  perpendicular,  therefore, 
passes  through  the  vertex,  and  the  two  sides  opposite  the 
equal  angles  of  the  triangle  are  equal. 


3.  Theorem.    In  a  triangle  ?/•////    inn-i/im/  <ttif/les  the   side 
oppotsitt    tli,'  greater  of  tw<>  «tti//<'8  in  greater  than  the  Me 
opposite  the  smaller;  and  conversely,  if  the  sides  of  a  triangle 
are  unequal  the  opposite  angles  art   unequal,  <///»/  the  greater 
angle  lies  oppo*/'f,   /•//,  ;ir>'<it,  /•  Me. 

4.  Theorem.    If  two  triangles  have  two  sides  of  one  equal, 
respectively,  to  two  sides  of  the  other,  but  the  included  angle 
of  the  first  greater  than  the  included  angle  of  the  second,  the 
third  side  of  the  first  is  greater  than  the  third  Me  of  the 
second  ;  and  conversely,  if  two  triangles  have  two  sides  of 


8 


NON-EUCLIDEAN  GEOMETRY 


- 


one  equal,  respectively,  to  two  sides  of  the  other,  but  the  third 
side  of  the  first  greater  than  the  third  side  of  the  second,  the 
angle  opposite  the  third  side  of  the  first  is.  greater  than  the 
angle  opposite  the  third  side  of  the  second. 

5,  Theorem.    The  sum  of  two  lines  drawn  from  any  point 
to  the  extremities  of  a  straight  line  is  greater  than  the  sum  of 
two  lines  similarly  drawn  but  included  by  them. 

6.  Theorem.    Through  any  point  one  perpendicular  only 
can  be  drawn  to  a  straight  line. 


D\ 


Proof.  Let  P'  be  the  position  which  P  would  take  if  the 
plane  were  revolved  about  AB  into  coincidence  with  itself. 
If  we  could  have  two  perpendiculars,  PC  and  PD,  from  P  to 
AB,  then  CP1  and  DP'  would  be  continuations  of  these  lines 
and  we  should  have  two  different  straight  lines  joining  P  and 
P'}  which  is  impossible. 

Corollary.  Two  right  triangles  are  equal  when  the  hypothe- 
nuse  and  an  acute  angle  of  one  are  equal,  respectively,  to  the 
hypothenuse  and  an  acute  angle  of  the  other. 

7.  Theorem.  The  perpendicular  is  the  shortest  line  that  can 
be  drawn  from  a  point  to  a  straight  line. 

Corollary.  In  a  right  triangle  the  hypothenuse  is  greater 
than  either  of  the  two  sides  about  the  right  angle. 


PROPOSITIONS   TRUE   FOR   RESTRICTED   FIGURES 


8,  Theorem.    If  oblique  lines  drawn  from  a  point  in  a  per- 
pendicular to  a  line  cut  off  unequal  distances  from  the  foot  of 
*he  perpendicular,  they  are  unequal,  and  the  more  remote 
is  the  greater ;  and  conversely,  if  two  oblique  lines  drawn 
from  a  point  in  a  perpendicular  are  unequal,  the  greater 
cuts  off  a  greater  distance  from  the  foot  of  the  perpendicular. 

9.  Theorem.    If  a  perpendicular  is  erected  at  the  middle 
point  of  a  straight  line,  any  point  not  in  the  perpendicular  is 
nearer  that  extremity  of  the  line  which  is  on  the  same  side  of 
the  perpendicular. 

Corollary.  Two  points  equidistant  from  the  extremities  of  a 
strait/lit  /hie  determine  a,  perpendicular  to  the  line  at  its  middle 
point. 

10.  Theorem.    Two  triangles  are  equal  when  they  have  three 
sides  of  one  equal,  respectively,  to  three  sides  of  the  other. 

11,  Theorem.    If  two  /////*•  ///  a  plane  erected  perpendicular 
to  a  third  are  unequal,   the  line  joining  their  extremities 
makes  unequal  angles  with  them,  the  greater  angle  with  the 
shorter  perpendicular. 


Proof.  Suppose  AC>  BD.  Produce  ED,  making  BE  — AC. 
Then  EEC  =  ACE.  But  BDC  >  EEC,  by  (1),  and  ACD  is  a, 
part  of  A  CE.  Therefore,  all  the  more  BDC  >  A  CD. 


10 


NON-EUCLIDEAN   GEOMETRY 


12.  Theorem.    If  the  two  angles  at  C  and  D  are  equal,  the 
perpendiculars  are  equal,  and  if  the  angles  are  unequal,  the 
perpendiculars  are  unequal,  and  the  longer  perpendicular 
makes  the  smaller  angle. 

13.  Theorem.    If  two  lines   are  perpendicular  to  a  third, 
points  on  either  equidistant  from  the  third  are  equidistant 
from  the  other. 

C       \        ,K  D 


H      . 

Proof.  Let  AB  and  CD  be  perpendicular  to  HK,  and  on  CD 
take  any  two  points,  C  and  D,  equidistant  from  K',  then  C 
and  D  will  be  equidistant  from  AB.  For  by  superposition  we 
can  make  D  fall  on  C,  and  then  DB  will  coincide  with  CA 

by  (6). 

The  following  propositions  of  Solid  Geometry  depend  di- 
rectly on  the  preceding  and  hold  true  at  least  for  any 
restricted  portion  of  space. 

14.  Theorem.    If  a  line  is  perpendicular  to  two  intersecting 
lines  at  their  intersection,  it  is  perpendicular  to  all  lines  of 
their  plane  passing  through  this  point. 

15.  Theorem.  "If  two  planes  are  perpendicular,  a  line  drawn 
in  one  perpendicular  to  their  intersection  is  perpendicular  to 
the  other,  and  a  line  drawn  through  any  point  of  one  perpen- 
dicular to  the  other  lies  entirely  in  the  first, 


THE   THREE    HYPOTHESES 


11 


16.  Theorem.    If  a  line  is  perpendicular  to  a  plane,  any 
plane  through  that  line  is  perpendicular  to  the  plane. 

17.  Theorem.    If  a  plane  is  perpendicular  to  each  of  two 
meeting  planes,  it  is  perpendicular  to  their  intersection. 


in.    THE   THREE    HYPOTHESES 

The  angles  at  the  extremities  of  two  equal  perpendiculars 

tlier  right  angles,  (icntr  anul.'s,  or  obtus<>  angles,  at  least 

for  restricted  figures.     We  shall  distinguish  the  three  cases 

.king  of  them  as  the  hypothesis  of  the  right  angle,  the 

liesis  of  the  acute  angle,  and  the  hypothesis  of  the  obtuse 

.  respectively. 

Theorem.    Tin1  line  joining  the  extremities  of  two  equal 
,'ti<li<'iitars  is,  at  least  for  any  restricted  portion  of  the 
.  equal  to,  greater  fJmii*  or  less  than  the  line  joining 
feet  in  the  three  hypotheses,  respectively. 

C  K  D 


A  H  B 

Proof.    Let  AC  and  BD  be  the  two  equal  perpendiculars  and 

\  a  third  perpendicular  erected  at  the  middle  point  of  AB. 

HA  and  KC  are  perpendicular  to  HK,  and  KC  is  equal 

to,  greater  than,  op  less  than  HA,  according  as  the  angle  at  C 

.s  equal  to,  less  than,  or  greater  than  the  angle  at  A  (II,  12). 

Hence,  CD,  the  double  of  KC,  is  equal  to,  greater  than,  or  less 

AB  in  the  three  hypotheses,  respectively. 


12 


NON-EUCLIDEAN   GEOMETRY 


Conversely,  if  CD  is  given  equal  to,  greater  than,  or  less 
than  AB,  there  is  established  for  this  figure  the  first,  second, 
or  third  hypothesis,  respectively. 

Corollary.  If  a  quadrilateral  has  three  right  angles,  the  sides 
adjacent  to  the  fourth  angle  are  equal  to,  greater  than,  or  less 
than  the  sides  opposite  them,  according  as  the  fourth  angle  is 
right,  acute,  or  obtuse. 

2.  Theorem.  If  the  hypothesis  of  a  right  angle  is  true  in  a 
single  case  in  any  restricted  portion  of  the  plane,  it  holds 
true  in  every  case  and  throughout  the  entire  plane. 

C'  D' 


Proof.  We^  have  now  a  rectangle ;  that  is,  a  quadrilateral 
with  four  right  angles.  By  the  corollary  to  the  last  propo- 
sition, its  opposite  sides  are  equal.  Equal  rectangles  can  be 
placed  together  so  as  to  form  a  rectangle  whose  sides  shall  be 
any  given  multiples  of  the  corresponding  sides  of  the  given 
rectangle. 

Now  let  A'B'  be  any  given  line  and  A'C'  and  B'D'  two  equal 
lines  perpendicular  to  A'B'  at  its  extremities.  Divide  A'C1,  if 
necessary,  into  a  number  of  equal  parts  so  that  one  of  these 
parts  shall  be  less  than  AC,  and  on  AC  and  BD  lay  off  AM 
and  BN  equal  to  one  of  these  parts,  and  draw  MN.  ABNM 
is  a  rectangle ;  for  otherwise  MN  would  be  greater  than  or 


THE   THREE    HYPOTHESES 


13 


less  than  A  B  and  CD,  and  the  angles  at  M  and  N  would  all  be 
acute  angles  or  all  obtuse  angles,  which  is  impossible,  since 
their  sum  is  exactly  four  right  angles.  Again,  divide  A'B' 
into  a  sufficient  number  of  equal  parts,  lay  off  one  of  these  parts 
on  .1  H  and  on  MN,  and  form  the  rectangle  APQM.  Rectangles 
equal  to  this  can  be  placed  together  so  as  exactly  to  cover  the 
figure  A  'B'D'C1,  which  must  therefore  itself  be  a  rectangle. 

3.  Theorem.  If  the  hypothesis  of  the  acute  angle  or  the 
hypothesis  of  the  obtuse  angle  holds  true  in  a  single  case 
wit  bin  a  restricted  portion  of  the  plane,  the  same  hypothesis 
hold*  true  for  every  case  within  any  such  portion  of  the  plane. 

C  K  D 


Proof.  Let  CD  move  along  A  C  and  BD,  always  cutting  off 
equal  distances  on  these  two  lines ;  or,  again,  let  A  C  and  BD 
move  along  on  the  line  AB  towards  HK  or  away  from  HK, 
always  remaining  perpendicular  to  AB  and  their  feet  always 
at  equal  distances  from  H.  The  angles  at  C  and  D  vary 
continuously  and  must  therefore  remain  acute  or  obtuse,  as 
the  case  may  be,  or  at  some  point  become  right  angles.  There 
would  then  be  established  the  hypothesis  of  the  right  angle, 
and  the  hypothesis  of  the  acute  angle  or  of  the  obtuse  angle 
could  not  exist  even  in  the  single  case  supposed. 

The  angles  at  C  and  D  could  not  become  zero  nor  180°  in  a 
restricted  portion  of  the  plane ;  for  then  the  three  lines  A  C, 
CD,  and  BD  would  be  one  and  the  same  straight  line. 


14 


NON-EUCLIDEAN    GEOMETRY 


4.  Theorem.  The  sum  of  the  angles  of  a  triangle,  at  least 
in  any  restricted  portion  of  the  plane,  is  equal  to,  less  than, 
or  greater  than  two  right  angles,  in  the  three  hypotheses, 

respectively. 

A 


FIG.  1. 


D  C 

FIG.  2. 


Proof.  Given  any  right  triangle,  ABD  (Fig.  1),  with  right 
angle  at  B,  draw  AC  perpendicular  to  AB  and  equal  to  BD. 
In  the  triangles  ADC  and  DAB,  AC  =  BD  and'  AD  is  common, 
but  DC  is  equal  to,  greater  than,  or  less  than  AB  in  the  three 
hypotheses,  respectively.  Therefore,  DAC  is  equal  to, greater 
than,  or  less  than  ADB  in  the  three  hypotheses,  respectively 
(II,  4).  Adding  BAD  to  both  of  these  angles,  we  have  ADB 
-{•BAD  equal  to,  less  than,  or  greater  than  the  right  angle 
BAG. 

Now  at  least  two  angles  of  any  restricted  triangle  are  acute. 
The  perpendicular,  therefore,  from  the  vertex  of  the  third 
angle  upon  its  opposite  side  will  meet  this  side  within  the 
triangle  and  divide  the  triangle  into  two  right  triangles. 
Therefore,  in  any  restricted  triangle  the  sum  of  the  angles 
is  equal  to,  less  than,  or  greater  than  two  right  angles  in  the 
three  hypotheses,  respectively. 

We  will  call  the  amount  by  which  the  angle-sum  of  a  tri- 
angle exceeds  two  right  angles  its  excess.  The  excess  of  a 
polygon  of  n  sides  is  the  amount  by  which  the  sum  of  its 
angles  exceeds  n  —  2  times  two  right  angles. 

It  will  not  change  the  excess  if  we  count  as  additional 
vertices  any  number  of  points  on  the  sides,  adding  to  the  sum 
of  the  angles  two  right  angles  for  each  of  these  points. 


THE   EXCESS  OF  POLYGONS  15 

5.  Theorem.  The  excess  of  a  polygon  is  equal  to  the  sum  of 
the  excesses  of  any  system  of  triangles  into  which  it  may  be 
divided. 


Proof.  If  we  divide  a  polygon  into  two  polygons  by  a  straight 
or  broken  line,  we  may  assume  that  the  two  points  where  it 
meets  the  boundary  are  vertices.  If  the  dividing  line  is  a 
broken  line,  broken  at  p  points,  the  total  sum  of  the  angles  of 
the  two  polygons  so  formed  will  be  equal  to  the  sum  of  the 
angles  of  the  original  polygon  plus  four  right  angles  for  each 
of  these  p  points,  and  the  sides  of  the  two  polygons  will  be 
the  sides  of  the  original  polygon,  together  with  the  p  +  1 
parts  into  which  the  dividing  line  is  separated  by  the  p  points, 
each  part  counted  twice. 

Let  N  be  the  sum  of  the  angles  of  the  original  polygon,  and 
n  the  number  of  its  sides.  Let  S'  and  n',  S"  and  n"  have  the 
same  meanings  for  the  two  polygons  into  which  it  is  divided. 
Then  we  have,  writing  R  for  right  angle, 

.S"+  S"  =  .S'  +  4y;/,\ 

and  n'  +  n"  =  n  +  2  (p  +  1). 

Therefore,       S'  -  2  (n'  -2)R  +  S"  -  2  (n"  -  2)  R 

=  S  +  ±pR  -2(n  +  2p-2)R 
=  S  -  2  (n  -  2)  R. 

Any  system  of  triangles  into  which  a  polygon  may  be  divided 
is  produced  by  a  sufficient  number  of  repetitions  of  the  above 
process.  Always  the  excess  of  the  polygon  is  equal  to  the 
sum  of  the  excesses  of  the  parts  into  which  it  is  divided. 


16  NON-EUCLIDEAN   GEOMETRY 

We  may  extend  the  notion  of  excess  and  apply  it  to  any 
combination  of  different  portions  of  the  plane  bounded  com- 
pletely by  straight  lines. 

Instead  of  considering  the  sum  of  the  angles  of  a  polygon, 
we  may  take  the  sum  of  the  exterior  angles.  The  amount  by 
which  this  sum  falls  short  of  four  right  angles  equals  the 
excess  of  the  polygon.  We  may  speak  of  it  as  the  deficiency 
of  the  exterior  angles. 


The  sum  of  the  exterior  angles  is  the  amount  by  which  we 
turn  in  going  completely  around  the  figure,  turning  at  each 
vertex  from  one  side  to  the  next.  If  we  are  considering  a 
combination  of  two  or  more  polygons,  we  must  traverse  the 
entire  boundary  and  so  as  always  to  have  the  area  considered 
on  one  side,  say  on  the  left. 

6.  Theorem.  The  excess  of  polygons  is  always  zero,  always 
negative,  or  always  positive. 

Proof.  We  know  that  this  theorem  is  true  of  restricted  tri- 
angles, but  any  finite  polygon  may  be  divided  into  a  finite 
number  of  such  triangles,  and  by  the  last  theorem  the  excess 
of  the  polygon  is  equal  to  the  sum  of  the  excesses  of  the 
triangles. 

When  the  excess  is  negative,  we  may  call  it  deficiency,  or 
speak  of  the  excess  of  the  exterior  angles. 

Corollary.  The  excess  of  a  polygon  is  numerically  greater 
than  the  excess  of  any  part  which  may  be  cut  off  from  it  by 
straight  lines,  except  in  the  first  hypothesis,  when  it  is  zero. 


AREA    OF   TRIANGLES 


17 


The  following  theorems  apply  to  the  second  and  third 
hypotheses. 

7.  Theorem.  By  diminishing  the  sides  of  a  triangle,  or  even 
one  side  while  the  other  two  remain  less  than  some  fixed 
length,  we  can  diminish  its  area  indefinitely,  and  the  sum 
of  its  angles  will  approach  two  right  angles  as  limit. 

C  D 


A  B 

Proof.  Lt-t  ABDC  be  a  quadrilateral  with  three  right  angles, 
A,  B,  and  ('.  A  perpendicular  moving  along  AB  will  con- 
stantly increase  or  decrease ;  for  if  it  could  increase  a  part 
of  the  way  and  decrease  a  part  of  the  way  there  would  be 
different  positions  where  the  perpendiculars  have  the  same 
length ;  a  perpendicular  midway  between  them  would  be  per- 
pendicular to  CD  also,  and  we  should  have  a  rectangle. 

Divide  AB  into  »  equal  parts,  and  draw  perpendiculars 
through  the  points  of  division.  The  quadrilateral  is  divided 
into  n  smaller  quadrilaterals,  which  can  be  applied  one  to 
another,  having  a  side  and  two  adjacent  right  angles  the  same 
in  all.  Beginning  at  the  end  where  the  perpendicular  is  the 
shortest,  each  quadrilateral  can  be  placed  entirely  within  the 

next.      Therefore,   the  first  has  its  area   less   than   -  th  of 

n 

the  area  of  the  original  quadrilateral,  and  its  deficiency  or 
excess  less  than  -  th  of  the  deficiency  or  excess  of  the  whole. 

Now  any  triangle  whose  sides  are  all  less  than  A  C  or  BD,  and 
one  of  whose  sides  is  less  than  one  of  the  subdivisions  of  AB, 


18  NON-EUCLIDEAN  GEOMETRY 

can  be  placed  entirely  within  this  smallest  quadrilateral.    Such 
a  triangle  has  its  area  and  its  deficiency  or  excess  less  than 

-  th  of  the  area  and  of  the  deficiency  or  excess  of  the  original 
n 

quadrilateral. 

Thus,  a  triangle  has  its  area  and  deficiency  or  excess  less 
than  any  assigned  area  and  deficiency  or  excess,  however 
small,  if  at  least  one  side  is  taken  sufficiently  small,  the 
other  two  sides  not  being  indefinitely  large. 

8.  Theorem.  Two  triangles  having  the  same  deficiency  or 
excess  have  the  same  area. 


Proof.  Let  A  OB  and  A  'OB1  have  the  same  deficiency  or  excess 
and  an  angle  of  one  equal  to  an  angle  of  the  other.  If  we  place 
them  together  so  that  the  equal  angles  coincide,  the  triangles 
will  coincide  and  be  entirely  equal,  or  there  will  be  a  quad- 
rilateral common  to  the  two,  and,  besides  this,  two  smaller 
triangles  having  an  angle  the  same  in  both  and  the  same 
deficiency  or  excess.  Putting  these  together,  we  find  again 
a  quadrilateral  common  to  both  and  a  third  pair  of  triangles 
having  an  angle  the  same  in  both  and  the  same  deficiency  or 
excess.  We  may  continue  this  process  indefinitely,  unless  we 
come  to  a  pair  of  triangles  which  coincide ;  for  at^rtrtime  can 
one  triangle  of  a  pair  be  contained  entirely  within  the  other, 
since  they  have  the  same  deficiency  or  excess. 


AREA   OF   TRIANGLES  19 

Let  so  denote  the  sum  of  the  sides  opposite  the  equal  angles 
of  the  first  two  triangles,  sa  the  sum  of  the  adjacent  sides,  and 
s  'a  that  portion  of  the  adjacent  sides  counted  twice,  which  is 
common  to  the  two  triangles  when  they  are  placed  together. 
Writing  o'  and  a'  for  the  second  pair  of  triangles,  o"  and  a" 
for  the  third  pair,  etc.,  we  have 

sa    =  s'd    -\-  so',  so    =  sa', 

sa'  =  s'a'  +  so",  so'  =  sa", 

.sV  =  s'a"  +  W.  etc.  =  .sV".  etc. 

.-.  <Sv,   =,'a   +.vV  +s'alv  H  ----  . 


Therefore,  the  expressions  N'//.  xV.  xV,  •••  diminish  indefi- 
nitely. Each  of  these  is  made  up  of  a  side  counted  twice 
from  one  and  a  side  counted  twice  from  the  other  of  a  pair  of 
triangles.  Thus,  if  we  carry  the  process  sufficiently  far,  the 
remaining  triangles  can  be  made  to  have  at  least  one  side  as 
small  as  we  please,  while  all  the  sides  diminish  and  are  less, 
for  example,  than  the  longest  of  the  sides  of  the  original 
triangles.  Therefore,  the  areas  of  the  remaining  triangles 
diminish  indefinitely,  and^as  the  difference  of  the  areas 
remains  the  same  for  each  pair  of  triangles,  this  difference 
must  be  zero.  The  triangles  of  each  pair  and,  in  particular, 
the  first  two  triangles  have  the  same  area. 


Let  A BC  and  DEF  have  the  same  deficiency  or  excess,  and 
suppose  AC<DF.      Produce  AC  to  C",  making  AC'  =  DF. 


20 


NON-EUCLIDEAN    GEOMETRY 


Then  there  is  some  point,  B!,  on  AB  between  A  and  B  such 
that  AB'C'  has  the  same  deficiency  or  excess  and  the  same 
area  as  ABC.  Place  AB'C'  upon  DEF  so  that  AC'  will  coin- 
cide with  DP,  and  let  DE'F  be  the  position  which  it  takes. 
If  the  triangles  do  not  coincide,  the  vertex  of  each  opposite 
the  common  side  DF  lies  outside  of  the  other.  The  two  tri- 
angles have  in  common  a  triangle,  say  D  OF,  and  besides  this 
there  remain  of  the  two  triangles  two  smaller  triangles  which 
have  one  angle  the  same  in  both  and  the  same  deficiency 
or  excess.  These  two  triangles,  and  therefore  the  original 
triangles,  have  the  same  area. 

9.  Theorem.    The  areas  of  any  two  triangles  are  propor- 
tional to  their  deficiencies  or  excesses. 


Proof.  A  triangle  may  be  divided  into  n  smaller  triangles 
having  equal  deficiencies  or  excesses  and  equal  areas  by  lines 
drawn  from  one  vertex  to  points  of  the  opposite  side.  Each  of 

these  triangles  has  for  its  deficiency  or  excess  -  th  of  the  defi- 

1 
ciency  or  excess  of  the  original  triangle,  and  for  its  area  -  th 

of  the  area  of  the  original  triangle. 

When  the  deficiencies  or  excesses  of  two  triangles  are  com- 
mensurable, say  in  the  ratio  m :  n,  we  can  divide  them  into 
m  and  n  smaller  triangles,  respectively,  all  having  the  same 
deficiency  or  excess  and  the  same  area.  The  areas  of  the 
given  triangles  will  therefore  be  in  the  same  ratio,  in  :  n. 


THE   SINE   AND   COSINE  21 

When  the  deficiencies  or  excesses  of  two  triangles,  A  and  B, 
are  not  commensurable,  we  may  divide  one  triangle,  A,  as 
above,  into  any  number  of  equivalent  parts,  and  take  parts 
equivalent  to  one  of  these  as  many  times  as  possible  from  the 


other,  leaving  a  remainder  which  has  a  deficiency  or  excess 
less  than  the  deficiency  or  excess  of  one  of  these  parts.  The 
portion  taken  from  the  second  triangle  forms  a  triangle,  B'. 
A  and  B'  have  their  areas  proportional  to  their  deficiencies  or 
excesses,  these  being  commensurable.  Now  increase  indefi- 
nitely the  number  of  parts  into  which  A  is  divided.  These 
parts  will  diminish  indefinitely,  and  the  remainder  when  we 
take  B1  from  B  will  diminish  indefinitely.  The  deficiency  or 
excess  and  the  area  of  B'  will  approach  those  of  B,  and  the 
triangles  .1  and  B  have  their  areas  and  their  deficiencies  or 
excesses  proportional. 

Corollary.  The  areas  of  two  polygons  are  to  each  oth< ,  at 
tli fir  deficiencies  or  excesses. 

10.  Theorem.  Given  a  right  triangle  with  a  fixed  angle ; 
if  the  sides  of  the  triangle  diminish  indefinitely,  the  ratio  of 
the  opposite  side  to  the  hypothenuse  and  the  ratio  of  the 
adjacent  side  to  the  hypothenuse  approach  as  limits  the  sine 
and  cosine  of  this  angle. 

Proof.  Lay  off  on  the  hypothenuse  any  number  of  equal 
lengths.  Through  the  points  of  division  Al}  At,  •  •  •  draw  per- 
pendiculars AtC-H  A^Ci,  -••  to  the  base,  and  to  these  lines 


22  NON-EUCLIDEAN   GEOMETRY 

produced  draw  perpendiculars  A2D1}  ABDZ,  •  •  -  each  from  the 
next  point  of  division  of  the  hypothenuse. 

The  triangles  OA1C1  and  A^A^D^  are  equal  (II,  6,  Cor.). 

therefore.  and  7^ 

OAZ  ^  OAi  OA2^  OAl 

the  upper  sign  being  for  the  second  hypothesis  and  the  lower 
sign  for  the  third  hypothesis. 


A 


JJo 


#1 


A* 


Pi< 


A  C  r !•"•»• —  j  ^   Cj 2      r — 

Assume 


OA,._   >  OA 


OAl 

PC, 
OAi 


and 

Since  OAr_l  =  (r  —  1)  OA^ 

and  also  =  (?•  —  1)  Ar_1Ar) 

(~\C*         f)f^  C*    A          C*         A 

,  .  ,  .      .  l/Vx.1    H^   l/V/r i  ,         OjYlii    O-. lJ"lj 1 

the  inequalities  -  — ^    and    - 

OA^OAr_^  OAl  '     OAr_i 

applied  to  the  angle  at  Ar_l  become 

AA~lDA~1%°rOA    ^    and    ~ALA$~OA=*' 

Ar_lA.r  U^ir_l  Ylr_j/lr        USir_l 

The  first  of  these  two  inequalities  may  be  written 

A          D  4/4 

Ar  —  1-ky  —  1  >  "dr  —  l^1)- 


THE   SINE   AND   COSINE 
Add  1  to  both  members, 


-OAT*^*^ 

But 


OAr  <     0^r_! 

Again,  Cr_lCr  ^  Dr_lAr. 

Hence,  from  the  second  inequality  above,  we  have 


or 


Add  1  to  both  members, 

<>rr    ^    <>A 


'    1  OC 

The  ratios  —  ^  and  —  being  less  than  1,  and  always  increas- 

ing or  always  decreasing  when  the  hypothenuse  decreases, 
approach  definite  limits.  These  limits  are  continuous  func- 
tions of  .1  ;  if  we  vary  the  angle  of  any  right  triangle  contin- 
uously, keeping  the  hypothenuse  some  fixed  length,  the  other 
two  sides  will  vary  continuously,  and  the  limits  of  their  ratios 
to  the  hypothenuse  must,  therefore,  vary  continuously. 

<  'ailing  the  limits  for  the  moment  sA  and  cA,  we  may  extend 
their  definition,  as  in  Trigonometry,  to  any  angles,  and  prove 
that  all  the  formulae  of  the  sine  and  cosine  hold  for  these 
functions.  Then  for  certain  angles,  30°,  45°,  60°,  we  can  prove 


24 


NON-EUCLIDEAN   GEOMETRY 


that  they  have  the  same  values  as  the  sine  and  cosine,  and 
their  values  for  all  other  angles  as  determined  from  their 
values  for  these  angles  will  be  the  same  as  the  corresponding 
values  of  the  sine  and  cosine. 

C 


Draw  a  perpendicular,  CF,  from  the  right  angle  C  to  the 
hypothenuse  AB.  The  angle  FOB  is  not  equal  to  A,  but  the 
difference,  being  proportional  to  the  difference  of  areas  of 
the  two  triangles  ABC  and  FBC,  diminishes  indefinitely  when 
the  sides  of  the  triangles  diminish.  From  the  relation 


^_  AC_       FB  -BC  _ 
~AC  ~AB  +  ~BC  ~AB  ~     ' 

we  have,  by  passing  to  the  limit, 

(cA)2  +  (sA)'2  =  1. 

Let  x  and  y  be  any  two  acute  angles,  and  draw  the  figures 
used  to  prove  the  formulae  for  the  sine  and  cosine  of  the  sum 
of  two  angles. 

The  angles  x  and  y  remaining  fixed,  we  can  imagine  all  of 
the  lines  to  decrease  indefinitely,  and  the  functions  sx,  ex,  sy, 
etc.,  are  the  limits  of  certain  ratios  of  these  lines. 

CA_  _  CE_  OB     EA_  BA_ 
OA     OB  OA     BA  OA' 

PC  _OD  OB  CD  BA 
~OA~~OE'O~A~'BA~OA 
oc  \ 

in  the  second  figure  ) . 

(JA  J 


THE    SINE   AND   COSINE 


25 


The  angles  at  M  are  equal  in  the  two  triangles  EMB  and 
CMO,  and  we  may  write 

CM      ME  4-  8       ME  ±  CM  4  8 


where  8  has  the  limit  zero. 


v° 


M 

The  angle  EAB,  or  x',  is  not  the  same  as  x,  but  differs  from 
./•  only  by  an  amount  which  is  proportional  to  the  difference  of 
the  areas  of  the  triangles  OMC  and  MAB,  and  which,  there- 
fore, diminishes  indefinitely.  Thus,  the  limits  of  sx'  and  ex' 
a  iv  .sv  and  ex. 

Finally,  as  the  two  triangles  A('\  and  Ji/>\  have  the  angle 
\  in  (•(•iiinion.  we  may  write 

/>.V       CW 4-8'      CJV  -  DN  +  8' 


AN  —  BN 


/;.v  ~       .i.v 
where  the  limit  of  8'  is  zero. 


Now  at  the  limits  our  identities  become 
s  (x  +  y)  =  sx  -  cy  +  ex  •  sy, 
c(x  H-  y)  =  ex  •  cy  —  sx  •  sij. 


26  NON-EUCLIDEAN    GEOMETRY 

By  induction,  these  formulae  are  proved  true  for  any  angles. 
Other  formulae  sufficient  for  calculating  the  values  of  these 
functions  from  their  values  for  30°,  45°,  and  60°  are  obtained 
from  these  two  by  algebraic  processes. 

If  the  sides  of  an  isosceles  right  triangle  diminish  indefi- 
nitely, the  angle  does  not  remain  fixed  but  approaches  45°, 
and  the  ratios  of  the  two  sides  to  the  hypothenuse  approach 
as  limits  s45°  and  c45°.  Therefore,  these  latter  are  equal, 
and  since  the  sum  of  their  squares  is  '1,  the  value  of  each  is 

— =9  the  same  as  the  value  of  the  sine  and  cosine  of  45°. 
V2 

Again,  bisect  an  equilateral  triangle  and  form  a  triangle  in 
which  the  hypothenuse  is  twice  one  of  the  sides.  When  the 
sides  diminish,  preserving  this  relation,  the  angles  approach 
30°  and  60°.  Therefore,  the  functions,  s  and  c,  of  these  angles 
have  values  which  are  the  same  as  the  corresponding  values  of 
the  sine  and  cosine  of  the  same  angles. 

Corollary.  When  any  plane  triangle  diminishes  indefinitely, 
the  relations  of  the  sides  and  angles  approach  those  of  the  sides 
and  angles  of  plane  triangles  in  the  ordinary  geometry  and 
trigonometry  with  which  we  are  familia  r. 

11.  Theorem.  Spherical  geometry  is  the  same  in  the  three 
hypotheses,  and  the  formulae  of  spherical  trigonometry  are 
exactly  those  of  the  ordinary  spherical  trigonometry. 

Proof.  On  a  sphere,  arcs  of  great  circles  are  proportional  to 
the  angles  which  they  subtend  at  the  centre,  and  angles  on  a 
sphere  are  the  same  as  the  diedral  angles  formed  by  the  planes 
of  the  great  circles  which  are  the  sides  of  the  angles.  Their 
relations  are  established  by  drawing  certain  plane  triangles 
which  may  be  made  as  small  as  we  please,  and  therefore  may 
be  assumed  to  be  like  the  plane  triangles  in  the  hypothesis 
of  a  right  angle.  These  relations  are,  therefore,  those  of  the 
ordinary  Spherical  Trigonometry. 


THE   THREE   GEOMETRIES  27 

The  three  hypotheses  give  rise  to  three  systems  of  Geometry, 
which  are  called  the  Parabolic,  the  Hyperbolic,  and  the  Elliptic 
Geometries.  They  are  also  called  the  Geometries  of  Euclid,  of 
Lobachevsky,  and  of  Kiemann.  The  following  considerations 
exhibit  some  of  their  chief  characteristics. 


C  D  D'  D' 

Given  PC  perpendicular  to  a  line,  CF;  on  the  latter  we  take 
CD  =  PC, 
DD'  =  Pit. 
.  D'D"  =  PD',  etc. 

Now  if  PC  is  sufficiently  short  (restricted),  it  is  shorter 
than  any  other  line  from  /'  to  the  line  C'F;  for  any  line  as 
short  as  PC  or  shorter  would  be  included  in  a  restricted  por- 
tion of  the  plane  about  the  point  P,  for  which  the  perpendicu- 
lar is  the  shortest  distance  from  the  point  to  the  line. 

Therefore,        PD  >  PC,          .'.  CD'  >  2  <  'It. 

PD'  >  PC,  etc.  ;      (  'It"  >  3  (  'D,  etc. 

Again,  in  the  three  hypotheses,  respectively, 
,  and 


DPD'  <  \  CPD,  CD'P  <  $  CDP, 

D'PD"  <  $DPD',  etc.,  CD"P  <  \  CD'P,  etc. 

At  P  we  have  a  series  of  angles.  In  the  first  hypothesis 
there  is  an  infinite  number  of  these  angles,  and  the  series 
forms  a  geometrical  progression  of  ratio  -J,  whose  value  is 


28  NON-EUCLIDEAN   GEOMETRY 

exactly  —  •    In  the  second  hypothesis  there  is  also  an  infinite 

number  of  these  angles,  and  the  terms  of  the  series  are  less 
than  the  terms  of  the  geometrical  progression.  The  value  of 

the  series  is.  therefore,  less  than  —  •     In  the  third  hypothesis 

we  have  a  series  whose  terms  are  greater  than  those  of  the 
geometrical  progression,  and,  therefore,  whether  the  series  is 

convergent  or  divergent,  we  can  get  more  than  —  by  taking  a 

sufficient  number  of  terms.  In  other  words,  we  can  get  a  right 
angle  or  more  than  a  right  angle  at  P  by  repeating  this  process 
a  certain  finite  number  of  times. 

The  angles  at  D,  D',  D",  •  •  •  are  exactly  equal  to  the  terms 
of  the  series  of  angles  at  P.  In  the  first  two  hypotheses  they 
approach  zero  as  a  limit. 

The  distances  CD,  CD',  CD",  •  •  -  increase  each  time  by  more 
than  a  definite  quantity,  CD-,  therefore,  if  we  repeat  the 
process  an  unlimited  number  of  times,  these  distances  will 
increase  beyond  all  limit.  Thus,  in  the  first  and  second 
hypotheses  we  prove  that  a  straight  line  must  be  of  infinite 
length. 

In  the  hypothesis  of  the  obtuse  angle  the  line  perpendicular 
to  PC  at  the  point  P  will  intersect  CF  in  a  point  at  a  certain 
finite  distance  from  C,  one  of  the  D's,  or  some  point  between. 
On  the  other  side  of  PC  this  same  perpendicular  will  intersect 
FC  produced  at  the  same  distance.  But  we  have  assumed  that 
two  different  straight  lines  cannot  intersect  in  two  points ; 
therefore,  for  us  the  third  hypothesis  cannot  be  true  unless 
the  straight  line  is  of  finite  length  returning  into  itself,  and 
these  two  points  are  one  and  the  same  point,  its  distance  from 
C  in  either  direction  being  one-half  the  entire  length  of  the 
line.  In  this  way,  however,  we  can  build  up  a  consistent 
Geometry  on  the  third  hypothesis,  and  this  Geometry  it  is 
which  is  called  the  Elliptic  Geometry. 


THE   THREE   GEOMETRIES 


29 


The  constructions  would  have  been  the  same,  and  very 
nearly  all  the  statements  would  have  been  the  same,  if  we  had 
taken  CD  any  arbitrary  length  on  CF. 

The  restriction  which  we  have  placed  upon  some  of 
the  propositions  of  this  chapter  is  necessary  in  the  third 
hypothesis. 

Thus,  in  the  proof  that  the  exterior  angle  of  a  triangle  is 
greater  than  the  opposite  interior  angle,  the  line  AD  drawn 
through  the  vertex  A  to  the  middle  point  7)  of  the  opposite 


side  was  produced  so  as  to  make  AE  =  2AD.  If  AD  were 
greater  than  half  the  entire  length  of  the  straight  line  deter- 
mined by  A  and  D,  this  would  bring  the  point  E  past  the  point 
A}  and  the  angle  CBE,  which  is  equal  to  the  angle  C,  instead 
of  being  a  part  of  the  exterior  angle  CBF,  becomes  greater 
than  this  exterior  angle. 

Again,  if  two  angles  of  a  triangle  are  equal  and  the  side 
between  them  is  just  an  entire  straight  line,  it  does  not  follow 
necessarily  that  the  opposite  sides  are  equal.  It  may  be  said, 


30  NON-EUCLIDEAN   GEOMETRY 

however,  that  the  opposite  sides  form  one  continuous  line,  and, 
therefore,  this  figure  is  not  strictly  a  triangle,  but  a  figure 


C 

somewhat  like  a  lune.      The  points  A  and  B  are  the  same 
point,  and  the  angles  A  and  B  are  vertical  angles. 

Finally,  though  we  assume  that  the  shortest  path  between 
two  points  is  a  straight  line,  it  is  not  always  true  that  a 
straight  line  drawn  between  two  points  is  the  shortest  path 
between  them.  We  can  pass  from  one  point  to  another  in 
two  ways  on  a  straight  line ;  namely,  over  each  of  the  two 
parts  into  which  the  two  points  divide  the  line  determined  by 
them.  One  of  these  parts  will  usually  be  shorter  than  the 
other,  and  the  longer  part  will  be  longer  than  some  paths 
along  broken  lines  or  curved  lines. 

When,  however,  the  straight  line  is  of  infinite  length,  that 
is,  in  the  hypothesis  of  the  right  angle  and  in  the  hypothesis 
of  the  acute  angle,  all  the  propositions  of  this  chapter  hold 
without  restriction. 

The  Euclidean  Geometry  is  familiar  to  all.  We  will  now 
make  a  detailed  study  of  the  Geometry  of  Lobachevsky,  and 
then  take  up  in  the  same  way  the  Elliptic  Geometry.. 


CHAPTER    II 
THE   HYPERBOLIC  GEOMETRY 

\V  K  have  now  the  hypothesis  of  the  acute  angle.  Two  lines 
in  a  plane  perpendicular  to  a  third  diverge  on  either  side  of 
their  common  perpendicular.  The  sum  of  the  angles  of  a 
triangle  is  less  than  two  right  angles,  and  the  propositions 
of  the  last  chapter  hold  without  restriction. 


I.    PARALLEL   LINES 

From  any  point,  P,  draw  a  perpendicular,  PC,  to  a  given 
lint'.  I/;,  and  let  PD  be  any  other  line  from  P  meeting  CB 
in  /A  If  D  move  off  indefinitely  on  CB,  the  line  PD  will 
approarh  a  limiting  position  /'/•„'. 

P 


PE  is  said  to  be  parallel  to  CB  at  P.  PE  makes  with  PC 
an  angle,  CPE,  which  is  called  the  angle  of  parallelism  for 
tho  perpendicular  distance  PC,  It  is  less  than  a  right  angle 
by  an  amount  which  is  the  limit  of  the  deficiency  of  the  tri- 
angle PCD.  On  the  other  side  of  PC  we  can  find  another 
line  parallel  to  CA  and  making  with  PC  the  same  angle  of 
parallelism.  We  say  that  PE  is  parallel  to  AB  towards  that 
part  which  is  on  the  same  side  of  PC  with  PE.  Thus,  at  any 

31 


32  NON-EUCLIDEAN  GEOMETRY 

point  there  are  two  parallels  to  a  line,  but  only  one  towards 
one  part  of  the  line.  Lines  through  P  which  make  with  PC 
an  angle  greater  than  the  angle  of  parallelism  and  less  than 
its  supplement  do  not  meet  AB  at  all.  We  write  II  (p)  to 
denote  the  angle  of  parallelism  for  a  perpendicular  distance,  p. 

1.  Theorem.    A  straight  line  maintains  its  parallelism  at 
all  points. 

A -^ v 

B 


H  D 

Let  AB  be  parallel  to  CD  at  E  and  let  F  be  any  other  point 
of  AB  on  either  side  of  E,  to  prove  that  AB  is  parallel  to  CD 
at  F. 

Proof.  To  H,  on  CD,  draw  EH  and  FH.  If  H  move  off 
indefinitely  on  CD,  these  two  lines  will  approach  positions  of 
parallelism  with  CD.  But  the  limiting  position  of  EH  is  the 
line  AB  passing  through  F,  and  if  the  limiting  position  of  FH 
were  some  other  line,  FK,  F  would  be  the  limiting  position  of 
If)  the  intersection  of  EH  and  FH. 

2.  Theorem.  If  one  line  is  parallel  to  another,  the  second 
is  parallel  to  the  first. 

Given  AB  parallel  to  CD,  to  prove  that  CD  is  parallel  to  AB. 

Proof.  Draw  AC  perpendicular  to  CD.  The  angle  CAB 
will  be  acute;  therefore,  the  perpendicular  CE  from  C  to  AB 
must  fall  on  that  side  of  A  towards  which  the  line  AB  is 
parallel  to  CD  (Chap.  I,  II,  1).  The  angle  ECD  is  then  acute 
and  less  than  CEB,  which  is  a  right  angle.  That  is,  we  have 
CAB  <  A  CD,  and  CEB  >  ECD. 


PARALLEL  LINES 


33 


If  the  line  CE  revolve  about  the  point  C  to  the  position  of 
' 'A.  the  angle  at  E  will  decrease  to  the  angle  A,  and  the  angle 
at  C  will  increase  to  a  right  angle.  There  will  be  some  posi- 
tion, say  CF,  where  these  two  angles  become  equal ;  that  is, 

CFB  =  FCD. 


FE 


Draw  MN  perpendicular  to  CF  at  its  middle  point  and 
revolve  the  figure  about  MX  as  an  axis.  CD  will  fall  upon 
the  original  position  of  AB,  and  AB  will  fall  upon  the  original 
position  of  CD.  Therefore,  CD  is  parallel  to  A  B. 

Corollary.    FB  and  CD  are  both  parallel  to  MN. 

F 


Proof.  FB  and  CD  are  symmetrically  situated  with  respect 
to  MN,  and  cannot  intersect  MN  since  they  do  not  intersect 
each  other.  Draw  FH  to  H,  on  CD,  intersecting  MN  in  K. 
If  H  move  off  indefinitely  on  CD,  FH  will  approach  the  posi- 
tion of  FB  as  a  limit.  Now  K  cannot  move  off  indefinitely 
before  H  does,  for  FK  <  FH.  But  again,  when  H  moves  off 
indefinitely,  K  cannot  approach  some  limiting  position  at  a 


34  NON-EUCLIDEAN   GEOMETRY 

finite  distance  on  MN;  for  FB,  and  therefore  CD,  would  then 
intersect  MN  and  each  other  at  this  point.  Therefore,  H  and 
K  must  move  off  together,  and  the  limiting  position  of  FH 
must  be  at  the  same  time  parallel  to  CD  and  MN. 

In  the  same  way  we  can  prove  that  any  line  lying  in  a 
plane  between  two  parallels  must  intersect  one  of  them  or  be 
parallel  to  both. 

3.  Theorem.  Two  lines  parallel  to  a  third  towards  the  same 
part  of  the  third  are  parallel  to  each  other. 


First,  when  they  are  all  in  the  same  plane. 

Let  AB  and  EF  be  parallel  to  CD,  to  prove  that  they  are 
parallel  to  each  other. 

Proof.  Suppose  AB  lies  between  the  other  two.  To  H,  any 
point  on  CD,  draw  AH  and  EH,  and  let  K  be  the  point  where 
EH  intersects  AB.  As  H  moves  off  indefinitely  on  CD,  AH 
and  EH  approach  as  limiting  positions  AB  and  EF.  Now  A' 
cannot  move  off  indefinitely  before  H  does,  for  EK  <  EH. 
But  again,  when  H  moves  off  indefinitely,  K  cannot  approach 
some  limiting  position  at  a  finite  distance  on  AB ;  for  this  point 
would  be  the  intersection  of  AB  and  EF,  and  the  limiting 
position  of  H,  whereas  H  moves  off  indefinitely  on  CD.  There- 
fore, H  and  K  must  move  off  together,  and  the  limiting  posi- 
tion of  EH  must  be  at  the  same  time  parallel  to  CD  and  AB. 


PARALLEL   LINES 


35 


If  AB,  lying  between  the  other  two,  is  given  parallel  to  CD 
and  EF,  EF  must  be  parallel  to  CD',  for  a  line  through  E 
parallel  to  CD  would  be  parallel  to  AB,  and  only  one  line  can 
be  drawn  through  E  parallel  to  A  B  towards  the  same  part. 


i> 


Second,  when  the  lines  are  not  all  in  the  same  plane. 

Let  AB  and  CD  be  two  parallel  lines  and  let  E  be  any  point 
not  in  their  plane. 

Proof.  ToHonCDdTSiw  AHandEH.  As  H  moves  off  indefi- 
nitely, AH  approaches  the  position  of  AB,  and  the  plane  E  I  // 
the  position  of  the  plane  EAB.  Therefore,  the  limiting  posi- 
tion of  EH  is  the  intersection  of  the  planes  ECD  and  E.  1  /;. 
The  intersection  of  these  planes  is,  then,  parallel  to  CD,  and 
in  the  same  way  we  prove  that  it  is  parallel  to  AB. 

Now,  if  EF  is  given  as  parallel  to  one  of  these  two  lines 
towards  the  part  towards  which  they  are  parallel,  it  must  be 
the  intersection  of  the  two  planes  determined  by  them  and 
the  point  E,  and  therefore  parallel  to  the  other  line  also. 

4.  Theorem.    Parallel  lines  continually  approach  each  other. 

Let  AB  and  CD  be  parallel,  and  from  A  and  B,  any  points 
on  AB,  drop  perpendiculars  AC  and  BD  to  CD.  Supposing 
that  B  lies  beyond  A  in  the  direction  of  parallelism,  we  are 
to  prove  that  BD  <  AC. 

Proof.  At  H,  the  middle  point  of  CD,  erect  a  perpendicular 
meeting  AB  in  K.  The  angle  BKH  is  an  acute  angle,  and  the 


36 


NON-EUCLIDEAN   GEOMETRY 


angle  AKH  is  an  obtuse  angle.     Therefore,  a  perpendicular  to 
HK  at  K  must  meet  CA  in  some  point,  E,  between  C  and  A 


D 


C  H 

and  DB  produced  in  some  point,  F,  beyond  B. 
(Chap.  I,  I,  12)  ;   therefore,  DB  <  CA. 


But  DF  =  CE 


Corollary.  If  AB  and  CD  are  parallel  and  AC  makes  equal 
angles  with  them  (like  FC  in  2  above),  then  EF,  cutting  off 
equal  distances  on  these  two  lines,  AE  =  CF,  on  the  side  towards 
which  they  are  parallel,  will  be  shorter  than  A  C. 


M 


H 


Proof.  MN,  perpendicular  to  AC  at  its  middle  point,  is 
parallel  to  AB  and  bisects  EF,  the  figure  being  symmetrical 
with  respect  to  MN.  EH,  the  half  of  EF,  is  less  than  AM, 
and  therefore  EF  is  less  than  A  C. 

5.  Theorem.  As  the  perpendicular  distance  varies,  starting 
from  zero  and  increasing  indefinitely,  the  angle  of  parallelism 
decreases  from  a  right  angle  to  zero. 

Proof.  In  the  first  place  the  angle  of  parallelism,  which  is 
acute  as  long  as  the  perpendicular  distance  is  positive,  will  be 


THE   ANGLE   OF   PARALLELISM 


37 


made  to  differ  from  a  right  angle  by  less  than  any  assigned 
value  if  we  take  a  perpendicular  distance  sufficiently  small. 


For,  ADE  being  any  given  angle  as  near  a  right  angle  as 
we  please,  we  can  take  a  point,  Z,  on  DE  and  draw  LR  perpen- 
dicular to  DA  at  R.  The  angle  RDL  must  increase  to  become 
the  angle  of  parallelism  for  the  perpendicular  distance  RD. 


Now  let  p  be  the  length  of  a  given  perpendicular  PM,  and 
let  a  be  the  amount  by  which  its  angle  of  parallelism  differs 

7T 

from  —  ;  that  is,  say 


I'M,  being  perpendicular  i<>  M.\,  and  //  any  point  on  MN,  the 
angle  MPH  approaches  as  a  limit  the  angle  of  parallelism, 
II  (  p),  when  H  moves  off  indefinitely  on  MN.  The  line  PH 
meets  the  line  MX  as  long  as  MPH  <  n  (y>),  and  by  taking 
MTH  sufficiently  near  II  (p),  but  less,  we  can  make  the  angle 
MIIP  as  small  as  we  please  (see  p.  27). 

In  figure  on  page  38,  let  A  C  be  perpendicular  to  AB,  D  being 
any  point  on  A  C  and  DE  parallel  to  AB.  Draw  DK  beyond  DE, 
making  .with  DE  an  angle,  EDK  =  U  (p),  and  make  DK  =  p. 
TF,  perpendicular  to  DK  at  K,  will  be  parallel  to  DE  and  AB. 


38 


NON-EUCLIDEAN   GEOMETRY 


By  placing  PMN  of  the  last  figure  upon  DKT,  we  see  that 
DC  will  meet  KT  in  a  point,  G  if 

KDC  <  H  O), 

that  is,  if  ADE>2a. 

Then  in  the  right  triangle  DKG, 


Starting  from  the  point  G,  we  can  repeat  this  construction, 
and  each  time  we  subtract  from  the  angle  of  parallelism  an 
amount  greater  than  a.  We  can  continue  this  process  until 
the  angle  of  parallelism  becomes  equal  to  or  less  than  2  a. 

If  the  point  D  move  along  AC,  DE  remaining  constantly 
parallel  to  AB,  the  angle  at  D  will  constantly  diminish,  and 
by  letting  D  move  sufficiently  far  on  A  C  we  can  reach  a  point 
where  this  angle  becomes  equal  to  or  less  than  2  a. 

Suppose  D  is  at  the  point  where  the  angle  of  parallelism  is 
just  2  a.  Then,  if  we  draw  DK  and  TF  as  before,  KT  will  be 


PERPENDICULARS   IX   A   TRIANGLE 


39 


parallel  to  DC.  All  the  parallels  to  AB  lying  between  AB 
and  this  position  of  TF  meet  A  C,  and  as  the  parallel  moves 
towards  this  position  of  TF,  the  angle  of  parallelism  at  D 
approaches  zero,  and  the  point  D  moves  off  indefinitely. 


For  an  obtuse  angle  we  may  take^>  negative,  and  we  have 


6.  Theorem.  The  perpendiculars  erected  at  the  middle 
points  of  the  sides  of  a  triangle  are  all  parallel  if  two  of 
tin  in  are  parallel. 


Let  .1,  B,  and  C  be  the  vertices  of  the  triangle,  and  D,  E, 
and  F,  respectively,  the  middle  points  of  the  opposite  sides. 
Suppose  the  perpendiculars  at  D  and  E  are  given  parallel,  to 
prove  that  the  perpendicular  at  F  is  parallel  to  them. 


40 !  NON-EUCLIDEAN   GEOMETRY 

Proof.  Draw  CM  through  C  parallel  to  the  two  given  par- 
allel perpendiculars.  CM  forms  with  the  two  sides  at  C  angles 

of  parallelism  II '.(  -  J   and  II  f  -  J ,  of  which  the  angle  at  C  is 

the  sum  or  difference  according  as  C  lies  between  the  given 
perpendiculars  or  on  the  same  side  of  both.  By  properly 
diminishing  these  angles  at  C,  keeping  the  lengths  of  CA 
and  CB  unchanged,  we  can  make  the  perpendiculars  at  their 
middle  points  D  and  E  intersect  CM,  and  therefore  each  other, 

at  any  distance  from  C  greater  than  —  and  greater  than  -• 

2i  2i 

Let  A'fi'C"  be  the  triangle  so  formed,  0  the  point  where  the 
two  given  perpendiculars  meet,  and  C'M'  the  line  through  0. 
In  the  triangle  A'B'C',  the  three  perpendiculars  meet  at  the 
point  0  (Chap.  I,  I,  5).  Now  we  can  let  0  move  off  on  C'M', 
the  construction  remaining  the  same.  That  is,  we  let  the 
lines  C'A'  and  C'B'  rotate  about  C"  without  changing  their 
lengths,  in  such  a  manner  that  the  three  perpendiculars  D'O, 
E'O,  and  F'O  shall  always  pass  through  0.  As  0  moves  off 

indefinitely,  the  angles  at  C'  approach  n  (  -  J  and  II  f  -  J  as 

limits,  and  the  three  perpendiculars  approach  positions  of 
parallelism  with  C'M1  and  with  each  other.  But  the  triangle 
A'B'C'  approaches  as  a  limit  a  triangle  which  is  equal  to  ABC, 
having  two  sides  and  the  included  angle  equal,  respectively,  to 
the  corresponding  parts  of  the  latter.  Therefore,  in  ABC  the 
three  perpendiculars  are  all  parallel. 

7.  Theorem.  Lines  which  do  not  intersect  and  are  not 
parallel  have  one  and  only  one  common  perpendicular. 

Proof.  Let  AB  and  CD  be  the  two  lines,  and  from  A,  any 
point  of  AB,  drop  AC  perpendicular  to  CD.  If  AC  is  not 
itself  the  common  perpendicular,  one  of  the  angles  which  it 
makes  with  A  B  will  be  acute.  Let  this  angle  be  on  the  side 


PERPENDICULARS  IN  A  TRIANGLE 


1 


towards  AB,  so  that  BAC  <  ~     Draw  AE  parallel  to  CD 

6 

on  this  same  side  of  A  C.  The  angle  EA  C  is  less  than  BA  C, 
since  AB  is  not  parallel  to  CD  and  does  not  intersect  it.  Let 
A  H  be  any  line  drawn  in  the  angle  EA  C,  intersecting  CD  at 
//.  If  H,  starting  from  the  position  of  C,  move  off  indefinitely 


OMB  7) 

on  the  line  CZ>,  the  angle  BA  H  will  decrease  from  the  magni- 
tude of  the  angle  BAC  to  the  angle  BA E.  The  angle  AHC 
will  decrease  indefinitely  from  the  magnitude  of  the  angle  at 
(\  which  is  a  right  angle  and  greater  than  BAC.  There  will 
be  some  position  for  which  BAH  =  AHC.  In  this  position 
the  line  NM  through  the  middle  point  of  AH  perpendicular 
to  one  of  the  two  given  lines  will  be  perpendicular  to  the 
other,  as  proved  in  Chap.  I,  I,  10. 

If  there  were  two  common  perpendiculars  we  should  have  a 
rectangle,  which  is  impossible  in  the  Hyperbolic  Geometry. 

8.  Theorem.    If  the  perpendiculars  erected  at  the  middle 


points  of  the  sides  of  a  triangle  do  not  meet  and  are  not 
llel,  they  are  all  perpendicular  to  a  certain  line. 


NON-EUCLIDEAN   GEOMETRY 


Proof.  We  can  draw  a  line,  AB,  that  will  be  perpendicular 
to  two  of  these  lines,  and  the  perpendiculars  from  the  three 
vertices  of  the  triangle  upon  this  line  will  be  equal,  by  Chap.  I, 
II,  13.  A  perpendicular  to  AB  erected  midway  between  any 
two  of  these  three  is  perpendicular  to  the  corresponding  side 
of  the  triangle  at  its  middle  point  (Chap.  1, 1, 11).  Thus,  all 
three  of  the  perpendiculars  erected  at  the  middle  points  of  the 
sides  of  the  triangle  are  perpendicular  to  AB. 

A  line  is  parallel  to  a  plane  if  it  is  parallel  to  its  projection 
on  the  plane. 

9.  Theorem.  A  line  may  be  drawn  perpendicular  to  a  plane 
and  parallel  to  any  line  not  in  the  plane. 

^B 
R,  D  A 


M 
X» 


x- 


N 


Proof.  Let  AB  be  the  given  line  and  MN  the  plane.  If  AB 
meets  the  plane  MN  at  a  point,  A,  we  take  on  its  projection  a 
length,  AC,  such  that  the  angle  at  A  equals  II (A C).  Then 
CD,  perpendicular  to  the  plane  at  C,  will  be  parallel  to  AB. 
In  the  same  way,  on  the  other  side  of  the  plane  a  perpendic- 
ular can  be  drawn  parallel  to  BA  produced. 

If  AB  does  not  meet  MN,  then  at  least  in  one  direction  it 
diverges  from  MN.  Through  H,  any  point  of  the  projection 
of  AB  on  the  plane,  we  can  draw  a  line,  HK,  parallel  to  AB 
towards  that  part  of  AB  which  diverges  from  MN,  and  then 
draw  CD  parallel  to  this  line  and  perpendicular  to  the  plane. 

Unless  AB  is  parallel  to  MN  it  will  meet  the  plane  at  some 
point,  or  the  plane  and  line  will  have  a  common  perpendicular, 
and  the  line  will  diverge  from  the  plane  in  both  directions. 


H(  >UNDARY-CURVES  43 

In  the  latter  case  there  are  two  perpendiculars  that  are  parallel 
to  the  line,  one  parallel  towards  each  part  of  the  line. 

Two  perpendiculars  cannot  be  parallel  towards  the  same 
part  of  a  line ;  for  then  they  would  be  parallel  to  each  other, 
and  two  lines  cannot  be  perpendicular  to  a  plain1  and  parallel 
to  each  other. 

II.    liolNDAKY-CURVES   AND   SURFACES,    AND    EQUI- 
DISTANT-CURVES  AND   SURFACES 

Having  given  the  line  ABy  at  its  extremity,  A,  we  take  any 
arbitrary  angle  and  produce  the  side  AC  so  that  the  perpen- 
dicular erected  at  its  middle  point  shall  be  parallel  to  A/:. 
The  locus  of  the  point  C  is  a  curve  which  is  called  oricycle,  or 
boundary-curve.  Mi  is  its  axis. 


From  their  definition  it  follows  that  all  boundary-curves 
are  equal,  and  the  boundary -curve  is  symmetrical  with  respect 
to  its  axis ;  if  revolved  through  two  right  angles  about  its 
axis,  it  will  coincide  with  itself. 

1.  Theorem.  Any  line  parallel  to  the  axis  of  a  boundary- 
curve  may  be  taken  for  axis. 

Let  AB  be  the  axis  and  rn  any  liin-  parallel  to  Ml,  to 
prove  that  CD  may  !»•  taken  as  axis. 


44 


NON-EUCLIDEAN  GEOMETRY 


Proof.  Draw  A  C ;  also  to  E,  any  other  point  on  the  curve, 
draw  AE  and  CE.  The  perpendiculars  erected  at  the  middle 
points  of  A  C  and  of  AE  are  parallel  to  AB  and  CD  and  to  each 


D 


other.  Therefore,  the  perpendicular  erected  at  the  middle 
point  of  CE,  the  third  side  of  the  triangle  A  CE,  is  parallel  to 
them  and  to  CD.  CD  then  may  be  taken  as  axis. 

Corollary.  The  boundary-curve  may  be  slid  along  on  itself 
without  altering  its  shape  ;  that  is,  it  has  a  constant  curvature. 

2.  Theorem.  Two  boundary -curves  having  a  common  set  of 
axes  cut  off  the  same  distance  on  each  of  the  axes,  and  the 
ratio  of  corresponding  arcs  depends  only  on  this  distance. 


Proof.    Take  any  two  axes  and  a  third  axis  bisecting  the 
arc  which  the  first  two  intercept  on  one  of  the  two  boundary- 


BOUNDARY-CURVES 


45 


curves.     By  revolving  the  figure  about  this  axis  we  show  that 
the  curves  cut  off  equal  distances  on  the  two  axes. 

Let  A  A ',  BB',  and  CC'  be  any  three  axes  of  the  two  boundary- 
curves  AB  and  A'B';  let  their  common  length  be  x  and  let 
them  intercept  arcs  s  and  t  on  AB,  s'  and  t'  on  .4'^'. 

When  s  =  t,  s'  =  t\  and,  in  general, 

s      .<?' 


as  we  prove,  first  when  s  and  t  are  commensurable,  and  then 
by  the  method  of  limits  when  they  are  incommensurable. 

The  ratio  —,  is,  therefore,  a  constant  for  the  given  value  of  x. 
s 

Write 


From  three  boundary-curves  having  the  same  set  of  axes, 

we  find  /(*  +  y)=/(s)/(y). 

This  property  is  characteristic  of  the  exponential  function 
whose  general  form  is  f(x)  =  e°**  Therefore,  -  =  eax,  the 
value  of  a  depending  on  the  unit  of  measure  (see  below  p.  76). 

*  Putting  y  =  x,  2  x,  •  •  •  (n  —  1)  x  in  succession,  we  find 

f(nx)  =  [/(z)]» 
for  positive  integer  values  of  n,  x  being  any  positive  quantity. 

Now 

and  this  is  the  rth  power  of  the  sth  root  of  the  first  member  of  the 
equation 


46 


NON-EUCLIDEAN   GEOMETRY 


3.  Theorem.  The  area  enclosed  by  two  boundary -curves 
having  the  same  axes  and  by  two  of  their  common  axes  is 
proportional  to  the  difference  of  the  intercepted  arcs. 


Proof.  Let  s  and  s'  be  the  lengths  of  the  intercepted  arcs, 
and  I  the  distance  measured  on  an  axis  between  them.  Let  t, 
t',  and  k  be  the'  corresponding  quantities  for  a  second  figure 
constructed  in  the  same  way. 

If  the  corresponding  lines  in  the  two  figures  are  all  equal, 
the  areas  are  equal,  for  they  can  be  made  to  coincide.  If 
only  k  =  /,  the  areas  are  to  each  other  as  corresponding  arcs, 
say  as  s' :  t',  proved  first  when  the  arcs  are  commensurable,  and 
then  by  the  method  of  limits  when  they  are  incommensurable. 


When  I  and  k  are  commensurable,  suppose 


in       n 


Thus,  assuming  that  f(x)  is  a  continuous  function  of  x,  we  have  proved 
that  for  all  real  positive  values  of  x  and  n 

f(nx)  =  [/(x)]«, 
and  if  we  put  x  for  n  and  1  for  x,  we  have 


We  will  write  /(I)  =  ea  ;  then 

/(x)  =  e»*. 


EQUIDISTANT-CURVES  47 

We  can  draw  a  series  of  boundary-curves  at  distances  equal  to 
a  on  the  axes  and  divide  the  areas  into  m  and  n  parts,  respec- 
tively. If  f  is  the  ratio  of  arcs  corresponding  to  the  distance 
a,  these  parts  will  be  proportional  to  the  quantities 

fi    t\r    fi't    .   .  /v»— i 

v  ,    f  r  j    i  i)  v  r 

The  two  areas  are  then  to  each  other  in  the  ratio 


But  .vV"  =  s  and  t'i»  =  f, 

so  that  this  is  the  same  as  the  ratio 

s  —  s' :  t  —  t'. 

When  I  and  k  are  incommensurable,  we  proceed  as  in  other 
similar  demonstrations. 

This  theorem  is  analogous  to  the  one  which  we  have  proved 
about  polygons :  the  area  is  proportional  to  the  amount  of 
rotation  in  excess  of  four  right  angles  in  going  around  the 
figure,  for  the  rate  of  rotation  in  going  along  a  boundary- 
curve  is  constant. 


The  locus  of  points  at  a  given  distance  from  a  straight  line 
is  a  curve  which  may  be  called  an  equidistant-curve.  The 
perpendiculars  from  the  different  points  of  this  curve  upon 
the  base  line  are  equal  and  may  be  called  axes  of  the  curve. 

An  equidistant-curve  fits  upon  itself  when  revolved  through 
two  right  angles  about  one  of  its  axes  or  when  slid  along  upon 
itself.  It  has  a  constant  curvature. 


48 


NON-EUCLIDEAN   GEOMETRY 


It  can  be  proved,  exactly  as  in  the  case  of  two  boundary- 
curves  having  the  same  set  of  axes,  that  arcs  on  an  equidis- 
tant-curve are  proportional  to  the  segments  cut  off  by  the 
axes  at  their  extremities  on  the  base  line  or  on  any  other 
equidistant-curve  having  the  same  set  of  axes. 

4.  Theorem.  The  boundary-curve  is  a  limiting  curve  between 
the  circle  and  the  equidistant- curve  ;  it  may  be  regarded  as  a 
circle  with  infinitely  large  radius,  or  as  an  equidistant-curve 
whose  base  line  is  infinitely  distant. 


F  C 


H 


Proof.  Take  a  line  of  given  length,  AB  =  2  a 'say,  making 
an  angle,  A,  with  a  fixed  line,  A  C.  Construct  another  angle  at 
B  equal  to  the  angle  A,  and  draw  a  perpendicular  to  AB  at  its 
middle  point,  D. 

If  the  angle  at  A  is  sufficiently  small,  we  have  an  isosceles 
triangle  with  AB  for  base,  and  its  vertex  at  a  point,  F,  on  A  C. 
With  F  as  centre,  we  can  draw  a  circle  through  the  points  A 
and  B.  Now  let  the  angle  at  A  gradually  increase,  the  rest 
of  the  figure  varying  so  as  to  keep  the  construction.  F  will 
move  off  indefinitely,  and  when  A  =  II  (a)  the  three  lines  AF, 
BF,  and  DF  will  become  parallel,  and  B  will  become  a  point 
on  the  boundary-curve  AB',  which  has  AC  for  axis. 

On  the  other  hand,  if  the  angle  at  A  were  taken  acute,  but 
greater  than  II  (a),  we  should  have  three  lines,  AE,  BH,  and 


BOUNDARY-CURVES   AS   LIMIT   CURVES  49 

DF,  perpendicular  to  a  line,  EH,  the  base  line  of  an  equidis- 
tant-curve through  the  points  A  and  B.  Now  let  the  angle  A 
gradually  decrease,  the  rest  of  the  figure  varying  so  as  to  pre- 
serve the  construction.  The  quadrilateral  ADFE,  having  three 
right  angles  and  the  fourth  angle  A  decreasing,  must  increase 
in  area.  We  get  this  same  movement  if  we  think  of  AD  and 
DF  remaining  fixed  in  the  plane  while  AE  revolves  about  .1, 
making  the  angle  A  decrease.  Thus  the  only  way  in  which 
the  area  of  the  quadrilateral  can  increase  is  for  EH  to  move 
off  along  on  AC  and  become  more  and  more  remote  from  A. 
When  A  becomes  equal  to  II  (a),  BH  and  DF  become  parallel 
to  AC.  and  />'  falls  on  the  boundary -curve  AB'. 

Calling  the  radius  of  a  circle  axis,  we  find  that  circles, 
boundary-curves,  and  equidistant-curves  have  many  properties 
in  common : 

The  perpendicular  erected  at  the  middle  point  of  any  chord 
is  an  axis.  In  particular,  a  tangent  is  perpendicular  to  the  axis 
drawn  from  its  point  of  contact.  These  are  curves  cutting 
at  right  angles  a  system  of  lines  through  a  point,  a  system 
of  parallel  lines,  and  the  perpendiculars  to  a  given  line, 
respectively. 

Two  of  these  curves  having  the  same  set  of  axes  cut  off 
equal  lengths  on  all  these  axes,  and  the  ratio  of  corresponding 
arcs  on  two  such  curves  is  a  constant  depending  only  on  the 
way  in  which  they  divide  the  axes. 

Three  points  determine  one  of  these  curves ;  that  is,  through 
any  three  points  not  in  a  straight  line  we  can. draw  a  curve 
which  shall  be  either  a  circle,  a  boundary-curve,  or  an  equi- 
distant-curve, and  through  any  three  points  only  one  such 
curve  can  be  drawn.  Any  triangle  may  be  inscribed  in  one 
and  only  one  of  these  curves. 

Each  of  these  curves  can  be  moved  on  itself  or  revolved  about 
any  axis  through  180°  into  coincidence  with  itself. 


50 


NON-EUCLIDEAN   GEOMETRY 


A  boundary-surface  or  orisphere  is  a  surface  generated  by 
the  revolution  of  a  boundary-curve  about  one  of  its  axes. 

5,  Theorem.    Any  line  parallel  to  the  axis  of  a  boundary- 
surface  may  be  regarded  as  axis. 


A 


K 


D' 


Let  A  A1  be  the  axis,  meeting  the  surface  at  .4,  and  BB'  a 
line  parallel  to  the  axis  through  any  other  point,  B,  of  the 
surface;  to  prove  that  BB'  may  be  regarded  as  axis. 

Proof.  Let  C  be  a  third  point  on  the  surface.  Draw  CC" 
through  C,  and  through  D,  E,  and  F,  the  middle  points  of  the 
sides  of  the  plane  triangle  ABC,  draw  DD',  EE',  and  FF'  all 
parallel  to  A  A'.  Finally,  let  OO'  be  parallel  to  these  lines  and 
perpendicular  to  the  plane  ABC.  The  projecting  planes  of 
the  other  parallels  all  pass  through  00'  (see  I,  9). 

Since  A  A'  is  axis  to  the  surface,  EE'  and  FF'  are  perpen- 
dicular to  AC  and  AB}  respectively.  Draw  FK  perpendicular 
to  the  plane  ABC  at  F.  It  will  lie  in  the  projecting  plane 
OFF'.  AB,  being  perpendicular  to  FF'  and  to  FK,  is  perpen- 


BOUNDARY-SURFACES  51 

dicular  to  this  plane,  OFF',  and  therefore  to  OF.  In  the  same 
way  we  prove  that  AC  is  perpendicular  to  OE.  Therefore, 
EC  is  perpendicular  to  OD  (Chap.  I,  I,  5).  But  OD  is  the 
intersection  of  the  plane  ABC  with  the  plane  ODD'.  Hence, 
IK'  is  perpendicular  to  this  plane  and  to  DD'  (Chap.  I,  II,  15). 

DD'  being  parallel  to  BB'  lies  in  the  plane  determined  by 
BB'  and  BC,  and  in  this  plane  only  one  perpendicular  can  be 
drawn  to  BC  at  its  middle  point.  Therefore,  if  we  pass  any 
plane  through  BB'  and  from  B  draw  a  chord  to  any  other 
point,  C,  o£  its  intersection  with  the  surface,  the  perpendicular 
in  this  plane  to  BC,  erected  at  the  middle  point  of  BC,  will 
be  parallel  to  BB'.  This  proves  that  the  section  is  a  boundary- 
curve,  having  BB'  for  axis,  and  that -the  surface  can  be  gener- 
ated by  the  revolution  of  such  a  boundary-curve  around  BB'. 

Therefore,  BB'  may  be  regarded  as  axis  of  the  surface. 

A  plane  passed  through  an  axis  of  a  boundary-surface  is 
called  a  principal  plane.  Every  principal  plane  cuts  the  sur- 
face in  a  boundary -curve.  Any  other  plane  cuts  the  surface  in 
a  circle ;  for  the  surface  may  be  regarded  as  a  surface  of  revo- 
lution having  for  axis  of  revolution  that  axis  which  is  perpen- 
dicular to  the  plane.  This  perpendicular  may  be  called  the  axis 
of  the  circle,  and  the  point  where  it  meets  the  surface,  the  pole 
of  the  circle.  The  pole  of  a  circle  on  a  boundary -surf  ace  is 
at  the  same  distance  from  all  the  points  of  the  circle,  distance 
being  measured  along  boundary -lines  on  the  surface. 

Any  two  boundary -surfaces  can  be  made  to  coincide,  and  a 
boundary-surface  can  be  moved  upon  itself,  any  point  to  the 
position  of  any  other  point,  and  any  boundary -curve  through 
the  first  point  to  the  position  of  any  boundary -curve  through 
the  second  point.  We  may  say  that  a  boundary-surface  has 
a  constant  curvature,  the  same  for  all  these  surfaces.  Figures 
on  a  boundary-surface  can  be  moved  about  or  put  upon  any 
other  boundary-surface  without  altering  their  shape  or  size. 


52  NON-EUCLIDEAN   GEOMETRY 

We  can  develop  a  Geometry  on  the  boundary-surface.  By 
line  we  mean  the  boundary-curve  in  which  the  surface  is  cut 
by  a  principal  plane.  The  angle  between  two  lines  is  the 
same  as  the  diedral  angle  between  the  two  principal  planes 
which  cut  out  the  lines  on  the  surface. 

6.  Theorem.  Geometry  on  the  boundary -surf ace  is  the  same 
as  the  ordinary  Euclidean  Plane  G-eometry. 

Proof.  On  two  boundary-surfaces  with  the  same  system  of 
parallel  lines  for  axes  corresponding  triangles  are  similar ;  that 
is,  corresponding  angles  are  equal,  having  the  same  measures 
as  the  diedral  angles  which  cut  them  out,  and  corresponding 
lines  are  proportional  by  (2).  But  we  can  place  these  figures 
on  the  same  surface ;  therefore,  on  one  boundary-surface  we 
can  have  similar  triangles.  Thus,  we  can  diminish  the  sides 
of  a  triangle  without  altering  their  ratios  or  the  angles.  We 
can  do  this  indefinitely ;  for  the  ratio  of  corresponding  lines 
on  the  two  surfaces,  being  expressed  by  the  function  eax  of  the 
distance  between  them,  can  be  made  as  large  as  we  please  by 
taking  x  sufficiently  large.  If  we  assume  that  figures  on  the 
boundary-surface  become  more  and  more  like  plane  figures 
when  we  diminish  indefinitely  their  size,  it  follows  that  a 
triangle  on  this  surface  approaches  more  and  more  the  form 
of  an  infinitesimal  plane  triangle,  for  which  the  sum  of  the 
angles  is  two  right  angles,  and  the  angles  and  sides  have 
the  same  relations  as  in  the  Euclidean  Plane  Geometry.  All 
the  formulae  of  Plane  Trigonometry  with  which  we  are  familiar 
hold,  then,  for  triangles  on  the  boundary-surface. 

On  the  boundary -surface  we  have  the  "hypothesis  of  the 
right  angle."  Eectangles  can  be  formed,  and  the  area  of  a 
rectangle  is  proportional  to  the  product  of  its  base  and  alti- 
tude, while  the  area  of  a  triangle  is  half  of  the  area  of  a 
rectangle  having  the  same  base  and  altitude. 


TRIGONOMETRICAL    FORMULAE 


53 


An  equidistant-surface  is  a  surface  generated  by  the  revo- 
lution of  an  equidistant-curve  about  one  of  its  axes.  It  is 
the  locus  of  points  at  a  given  perpendicular  distance  from  a 
plane.  Any  perpendicular  to  the  plane  may  be  regarded  as 
an  axis,  and  the  surface  is  a  surface  cutting  at  right  angles  a 
system  of  lines  perpendicular  to  the  plane.  The  surface  has 
a  constant  curvature,  fitting  upon  itself  in  any  position. 


III.    TRIGONOMETRICAL   FORMULAE 

1.  Let  ABC  be  a  plane  right  triangle.  Erect  A  A'  perpen- 
dicular to  its  plane  and  draw  BB'  and  CC'  parallel  to  A  A'. 
Draw  a  boundary-surface  through  A,  having  these  lines  for 
axes  and  forming  the  boundary-surface  triangle  AB"C".  Also 
construct  the  spherical  triangle  about  the  point  B. 


c6-n(a) 


The  angle  A  is  the  same  in  the  plane  triangle  and  in  the 
boundary -surf  ace  triangle.  The  planes  through  A  A'  are  per- 
pendicular to  ABC.  Hence,  the  spherical  triangle  has  a  right 
angle  at  the  vertex  which  lies  on  c,  and  EC  being  perpendic- 
ular to  CA  is  perpendicular  to  the  plane  of  CC'  and  A  A1. 
Therefore,  the  plane  BCC'  is  perpendicular  to  the  plane  ACC'j 


54  NON-EUCLIDEAN   GEOMETRY 

and  the  diedral  whose  edge  is  EC  has  for  plane  angle  the 
angle  ACC' =  H(b).  Since  the  boundary-surface  triangle  is 
right-angled  at  C",  the  angle  B",  or  what  is  the  same  thing,  the 
diedral  whose  edge  is  BB',  is  the  complement  of  the  angle  A. 

In  the  spherical  triangle  the  side  opposite  the  right  angle 
is  II  (a),  the  two  sides  about  the  right  angle  are  II  (c)  and  B, 
and  the  opposite  angles  are  II  (£)  and  90°  —  A. 

Applying  to  these  quantities  the  trigonometrical  formulae 
for  spherical  right  triangles,  we  get  at  once  the  relations  that 
connect  the  sides  and  angles  of  plane  right  triangles. 

Produce  to  quadrants  the  two  sides  about  the  angle  whose 
value  is  the  complement  of  A.  We  form  in  this  way  a  spher- 
ical right  triangle  in  which  the  side  opposite  the  right  angle 
is  the  complement  of  II  (c),  the  two  sides  about  the  right  angle 
are  the  complements  of  II  (a)  and  n  (&),  and  their  opposite 
angles  are  the  complements  of  B  and  A.  From  this  triangle 
we  deduce  the  following  rule  for  passing  from  the  formulae  of 
spherical  right  triangles  to  those  of  plane  triangles  : 

Interchange  the  two  angles  (or  the  two  sides)  and  everywhere 
use  the  complementary  function,  taking  the  corresponding 
angle  of  parallelism  for  the  sides. 

The  formulae  for  spherical  right  triangles  are 

sin  a  sin  b 

sin  A  =  —. sin  B  =  —  —  - 

sin  c  sin  c 

tan  b  tan  a 

cos  A  =  —   —  •  cos  B  — 

tan  c  tan  c 

tan  a  tan  b 

tan  A  =  — — -  •  tan  B  =  — 

sin  b  sm  a 

cos  B  cos  A 

sm  A  = •  sin  B  = 

cos  b  ,  .          cos  a 

cos  c  =  cos  a  cos  b. 
cos  c  =  cot  A  cot  B. 


TRIGONOMETRICAL   FORMULAE 


55 


From  these,  by  the  rule  given  on  the  previous  page,  we 
derive  the  following  formulae  for  plane  right  triangles  : 


cosB  = 


cos  n  (a) 


sin  B  = 


cot  n  ( 


cot  II  (a) 
cot  B  =  - 

cos  II  (b) 

sin  A 
cosB  = 


cos  A  = 


sin  A  = 


cot  A  = 


cos  A  = 


cos  n  (b) 

cos  n  (c) 

cot  II  (a) 


sin  n  (b)  sin  n  (a) 

sin  II  (c)  =  sin  II  (a)  sin  II  (b). 
sin  II  (c)  =  tan  A  tan  B.* 

We  can  obtain  the  formulae  for  oblique  plane  triangles  by 
dropping  a  perpendicular  from  one  vertex  upon  the  opposite 
side,  thus  forming  two  right  triangles. 


2.    Take  the  relation 


sin  II  (a)  = 


sin  B 
cos  A 


Let  p,  q,  and  r  be  the  sides  of  the  triangle  AB"C"  of  our 
last  demonstration  and  p',  y',  and  r  the  corresponding  sides 

*  We  can  arrange  the  parts  of  a  right  triangle  so  as  to  apply  Napier's 
rules ;  namely,  the  arrangement  would  be 


co-b 


56 


NON-EUCLIDEAN   GEOMETRY 


of  the  triangle  formed  in  the  same  way  on  a  boundary-surface 
tangent  to  the  plane  ABC  at  B. 


g 
COS  A  =  *• 


Now  <7  and  q'  are  corresponding  arcs 
on  two  boundary -curves  which  have  the 
same  set  of  parallel  lines  as  axes,  and 
their  distance  apart,  x,  is  the  distance 
from  a  boundary-curve  of  the  extremity 
of  a  tangent  of  arbitrary  length,  a.  Thus, 
we  have  for  corresponding  arcs 

s' 

-  =  sin  n  (a). 


3,  To  MN,  a  given  straight  line,  erect  a  perpendicular  at  a 
point,  O,  and  on  this  perpendicular  lay  off  OA  =  y  below  MN, 
and  OB  and  BP  each  equal  to  x  above  MN9  x  and  y  being  any 
arbitrary  lengths.  At  P  draw  PR  perpendicular  to  OP  and 


THE   ANGLE   OF   PARALLELISM 


57 


extending  towards  the  left,  and  through  B  draw  EF  making 
with  OP  an  angle  II  (z),  and  therefore  parallel  on  one  side  to 
ON  and  on  the  other  side  to  PR.  Finally,  draw  AK  and  AH, 
the  two  parallels  to  EF  through  A. 


At  the  point  .4  we  have  four  angles  of  parallelism  : 
CAK=CAH  =  U(AC), 


Therefore,  U(y)  =  U  (A  C)  +  BA  C, 

and  HO  +  2z)  =  n(.4C)-  BAG. 

Now  in  the  right  triangle  A  !',< 


cos  n  (y  +  a-)  = 


_  cos  n  (A  C) 


cosBAC 


or 


1  —  cos  II  (y  4-  x)  _  cos  BA  C  —  cos  II  (A  C) 
1  4-  cos  n  (y  4-  x)  ~~  cos^^C  4-  cos  II  (4  C) 


cos  £  [n  (^  C)  +  5-4  C]  cos 


58  NON-EUCLIDEAN  GEOMETRY 

whence, 


x)  =  tan  $U  (y)  tan  Jn(y  + 

tan  -J-  II  (aj)  is  then  a  function  of  #,  say  f(x)9  satisfying  the 
condition 


and  putting  successively  in  this  equation  y  +  x,  y  -\-2x,  etc., 
for  y,  we  may  add 

no;) 


II  (0)  =  —  and  tan  -J-  II  (0)  =  1  ;  therefore,  putting  y  =  0  in 

L 

the  first  and  last  of  all  these  fractions,  we  have 


or 


This  equation  is  characteristic  of  the  exponential  function.* 
II  (x)  being  an  acute  angle,  tan  £  II  (a;)  <  1  ;  therefore,  we  may 
write  /(I)  =  e~a>,  so  that  /(a;)  =  e~a>x.  a'  depends  on  the  unit 
of  measure  ;  we  will  take  the  unit  so  that  a'  =  1.  Finally, 
since  II  (—  x)  =  TT  —  II  (x), 

tan  £  H  (-  x)  =  cot  £  n  (a:)  =  [tan  ^  II  (x)]-  ^ 
That  is,  for  all  real  values  of  x 

tan  -i-  II  (ic)  =  e"*, 

*  See  footnote,  p.  45. 


THE   ANGLE   OF  PARALLELISM  59 

1  —  COS  II  (x) 

or  : — „  .  ^   /  =  cos  ix  +  i  sm  ix* 

sm  II  (x) 

*  i  stands  for  V—  1.  The  best  way  to  get  the  relations  between  the 
exponential  and  trigonometrical  functions  is  by  their  developments  in 
series :  X2  ~n 


cos*  =  l-|+|-. ..  +  (-!>» 
sinx  =  x-2L  +  ^-, +  ( 


These  series  are  convergent  for  all  values  of  x. 
Putting  ix  for  x,  we  have 


i.e.  ,  e*  —  cos  x  -j-  i  sin  x. 

Also  e~  IX  =  cos  x  —  i  sin  x. 

.-.  cosx  =  $(ef* 


Again,  putting  ix  for  x,  we  have 

ex  =  cos  ix  —  i  sin  ix, 
c~x  =  cos  ix  +  i  sin  ix  ; 
and  cos  ix  =  £(e*  +  e--'), 


sin  ix  = (e1  —  e-x). 

2il 

cos  ix  =  1  +  ^-,  +  ?-,  -f  •  • 


For  real  values  of  x,  cos  ix  and  —  are  real  and  positive,  and  vary 
from  1  to  co  as  x  varies  from  0  to  co. 

In  the  equation  cos2ix  +  sin2ix  =  1,  the  fii'st  term  is  real  and  positive 
for  real  values  of  x,  the  second  term  is  real  and  negative  ;  therefore,  sinix 
is  in  absolute  value  less  than  cos  ix,  and  tan  ix  is  in  absolute  value  less 
than  1.  tan  ix  varies  in  absolute  value  from  0  to  1  as  x  varies  from  0  to  w. 


60  NON-EUCLIDEAN   GEOMETRY 

Changing  the  sign  of  x,  we  have 

1  +  cos  H  (35) 

: —        .v      =  cos  ix  —  i  sin  ix, 

sin  II  (x) 

and,  adding  and  subtracting, 

1 

— — _,  .  =  cos ix, 
sin  n  (a) 

cot  II  (x)  =  —  i  sin  zee. 

The  nature  of  the  angle  of  parallelism  is,  therefore,  expressed 
by  the  equations 

sin  H(x)  = —> 

v  '      cos  tx 

tan  n  (x)  =  —. — — > 
sin  ix 

taxiix 

cosllfo)  =  — : — 
i 

4,  Substituting  in  the  formulae  of  plane  right  triangles,  we 
find  that  they  reduce  to  those  of  spherical  right  triangles  with 
ia,  ib,  and  ic  for  a,  b,  and  c,  respectively.  The  formulae  'of 
oblique  triangles  are  obtained  from  those  of  right  triangles 
in  the  same  way  as  on  the  sphere,  and  thus  all  the  formulae 
of  Plane  Trigonometry  are  obtained  from  those  of  Spherical 
Trigonometry  simply  by  making  this  change. 

As  fundamental  formulae  for  oblique  triangles  we  write 

sin  A  _  sinJS  _  sin  C 
sin  ia      sin  ib      sin  ic 

cos  ia  =      cos  ib  cos  ic  +  sin  ib  sin  ic  cos  A, 
cos  A  =  —  cos  B  cos  C  +  sin  B  sin  C  cos  ia. 
In  the  notation  of  the  Il-f unction,  these  are 

sin  A  tan  n  (a)  =  sin  B  tan  n  (b)  =  sin  C  tan  n  (c), 


INFINITESIMAL  TRIANGLES  61 

sinn  (ft)  sin  n(C)  =    _  cos        cog        cog  ^ 

sm  n  (a) 

,   sin  B  sin  C 

COS  yl  =  —  COS  B  COS  C  H : — „  .    . — 

sin  II  (a) 

5.  Since  for  very  small  values  of  x  we  have  approximately 

sin  ix  =  ixj 

x* 
cos  ix  =  1  4-  —  > 

tan  ix  =  to-, 
our  formulae  for  infinitesimal  triangles  reduce  to 

sin  A  _  sin£  _  sin  C 
a  b       t       c 

a2  =  b*  +  c*  —  2bc  cos  A, 
cos  .4  =  —  cos  (B  +  C). 

6.  Triangles  on  an  equidistant-surface  are  similar  to  their 
projections  on  the  base  plane ;  that  is,  they  have  the  same 
angles  and  their  sides  are .  proportional.     Thus  the  formulae 
of  Plane  Trigonometry   hold  for  any   equidistant-surface  if 
with  the  letters  representing  the  sides  we  put,  besides  i,  a 
constant  factor  depending  on  the  distance  of  the  surface  from 
the  plane. 


CHAPTER   III 

THE   ELLIPTIC  GEOMETRY 

IN  the  hypothesis  of  the  obtuse  angle  a  straight  line  is 
of  finite  length  and  returns  into  itself.  This  length  is  the 
same  for  all  lines,  since  any  two  lines  can  be  made  to  coin- 
cide. Two  straight  lines  always  intersect,  and  two  lines 
perpendicular  to  a  third  intersect  at  a  point  whose  distance 
from  the  third  on  either  line  is  half  the  entire  length  of  a 
straight  line. 


1.  A  straight  line  does  not  divide  the  plane.  Starting  from 
the  point  of  intersection  of  two  lines  and  passing  along  one  of 
them  a  certain  finite  distance,  we  come  to  the  intersection 
point  again  without  having  crossed  the  other  line.  Thus,  we 
can  pass  from  one  side  of  the  line  to  the  other  without  having 
crossed  it. 

There  is  one  point  through  which  pass  all  the  perpendiculars 
to  a  given  line.  It  is  called  the  pole  of  that  line,  and  the  line 
is  its  polar.  Its  distance  from  the  line  is  half  the  entire 
length  of  a  straight  line,  and  the  line  is  the  locus  of  points 
at  this  distance  from  its  pole.  Therefore,  if  the  pole  of  one 

62 


POLES  AND  POLARS  63 

line  lies  on  another,  the  pole  of  the  second  lies  on  the  first,  and 
the  intersection  of  two  lines  is  the  pole  of  the  line  joining 
their  poles. 

The  locus  of  points  at  a  given  distance  from  a  given  line  is 
a  circle  having  its  centre  at  the  pole  of  the  line.  The  straight 
line  is  a  limiting  form  of  a  circle  when  the  radius  becomes 
equal  to  half  the  entire  length  of  a  line. 

We  can  draw  three  lines,  each  perpendicular  to  the  other 
two,  forming  a  trirectangular  triangle.  Tt  is  also  a  self -polar 
triangle ;  each  vertex  is  the  pole  of  the  opposite  side. 

2,  All  the  perpendiculars  to  a  plane  in  space  meet  at  a 
point  which  is  the  pole  of  the  plane.  It  is  the  centre  of 
a  system  of  spheres  of  which  the  plane  is  a  limiting  form 
when  the  radius  becomes  equal  to  half  the  entire  length  of  a 
straight  line. 

Figures  on  a  plane  can  be  projected  from  similar  figures  on 
any  sphere  which  has  the  pole  of  the  plane  for  centre.  That 
is,  they  have  equal  angles  and  corresponding  sides  in  a  con- 
stant ratio  that  depends  only  on  the  radius  of  the  sphere. 
Two  corresponding  angles  are  equal,  because  they  are  the  same 
as  the  diedral  angles  formed  by  the  two  planes  through  the 
centre  of  the  sphere  which  cut  the  sphere  and  the  plane  in 
the  sides  of  the  angles.  Corresponding  lines  are  proportional ; 
for  if  two  arcs  on  the  sphere  are  equal,  their  projections  on  the 
plane  are  equal ;  and  that,  in  general,  two  arcs  have  the  same 
ratio  as  their  projections  on  the  plane  is  proved,  first  when 
they  are  commensurable,  and  by  the  method  of  limits  when 
they  are  incommensurable. 

Geometry  on  a  plane  is,  therefore,  like  Spherical  Geometry, 
but  the  plane  corresponds  to  only  half  a  sphere,  just  as  the 
diameters  of  a  sphere  correspond  to  the  points  of  half  the 
surface.  Indeed,  the  points  and  straight  lines  of  a  plane 
correspond  exactly  to  the  lines  and  planes  through  a  point, 


64  NON-EUCLIDEAN  GEOMETRY 

but  we  can  realize  the  correspondence  better  that  compares 
the  plane  with  the  surface  of  a  sphere.  If  we  can  imagine 
that  the  points  on  the  boundary  of  a  hemisphere  at  opposite 
extremities  of  diameters  are  coincident,  the  hemisphere  will 
correspond  to  the  elliptic  plane.  There  is  no  particular  line 
of  the  plane  that  plays  the  part  of  boundary.  All  lines  of 
the  plane  are  alike ;  the  plane  is  unbounded,  but  not  infinite 
in  extent. 

The  entire  straight  line  corresponds  to  a  semicircle.  We 
will  take  such  a  unit  for  measuring  length  that  the  entire 
length  of  a  line  shall  be  TT  ;  the  formulae  of  Spherical  Trigo- 
nometry will  then  apply  without  change  to  our  plane.  Dis- 
tances on  a  line  will  then  have  the  same  measure  as  the  angles 
which  they  subtend  at  the  pole  of  the  line,  and  the  angle 
between  two  lines  will  be  equal  to  the  distance  between  their 
poles.  The  distance  from  any  point  to  its  polar,  half  the 
entire  length  of  a  straight  line,  may  then  be  called  a  quadrant. 

We  can  form  a  self -polar  tetraedron  by  taking  three  mutually 
perpendicular  planes  and  the  plane  which  has  their  intersec- 
tion for  pole.  The  vertices  of  this  tetraedron  are  the  poles  of 
the  opposite  faces.  At  each  vertex  is  a  trirectangular  triedral, 
and  each  face  is  a  trirectangular  triangle. 

3.  Theorem.  All  the  planes  perpendicular  to  a  fixed  line 
intersect  in  another  fixed  line,  catted  its  polar  or  conjugate. 
The  relation  is  reciprocal,  and  all  the  points  of  either  line 
are  at  a  quadrant's  distance  from  all  the  points  of  the  other. 

Proof.  Let  the  two  planes  perpendicular  to  the  line  A  B  at 
H  and  K  intersect  in  CD.  Pass  a  plane  through  AB  and  R, 
any  point  of  CD.  This  plane  will  intersect  the  two  given 
planes  in  HR  and  KR.  HR  and  KR  are  perpendicular  to  AB ; 
therefore,  R  is  at  a  quadrant's  distance  from  H  and  K.  R  is 
then  the  pole  of  AB  in  the  plane  determined  by  AB  and  R, 


POLAR   LINES 


65 


and  is  at  a  quadrant's  distance  from  every  point  of  AB.  But 
R  is  any  point  of  CD ;  therefore,  any  point  of  either  line  is  at 
a  quadrant's  distance  from  each  point  of  the  other  line,  and  a 
point  which  is  at  a  quadrant's  distance  from  one  line  lies  in 


the  other  line.  Again,  any  point,  //,  of  AB,  being  at  a  quad- 
rant's distance  from  all  the  points  of  CD,  is  the  pole  of  CD  in 
the  plane  determined  by  it  and  CD.  Thus,  HR  and  KR  are 
both  perpendicular  to  CD,  and  the  plane  determined  by  AB 
and  R  is  perpendicular  to  CD. 

The  opposite  edges  of  a  self-polar  tetraedron  are  polar  lines. 

All  the  lines  which  intersect  a  given  line  at  right  angles 
intersect  its  polar  at  right  angles.  Therefore,  the  distances 
of  any  point  from  two  polar  lines  are  measured  on  the  same 
straight  line  and  are  together  equal  to  a  quadrant.  Two 
points  which  are  equidistant  from  one  line  are  equidistant 
from  its  polar. 

The  locus  of  points  which  are  at  a  given  distance  from  a 
fixed  line  is  a  surface  of  revolution  having  both  this  line  and 
its  polar  as  axes.  We  may  call  it  a  surface  of  double  revolu- 
tion. The  parallel  circles  about  one  axis  are  meridian  curves 
for  the  other  axis.  If  a  solid  body,  or,  we  may  say,  all  space, 
move  along  a  straight  line  without  rotating  about  it,  it  will 
rotate  about  the  conjugate  line  as  an  axis  without  sliding 


66 


NON-EUCLIDEAN   GEOMETRY 


along  it.  A  motion  along  a  straight  line  combined  with  a 
rotation  about  it  is  called  a  screw  motion.  A  screw  motion 
may  then  be  described  as  a  rotation  about  each  of  two  con- 
jugate lines  or  as  a  sliding  along  each  of  two  conjugate  lines. 

4.  Theorem.  In  the  elliptic  geometry  there  are  lines  not 
in  the  same  plane  which  have  an  infinite  number  of  common 
perpendiculars  and  are  everywhere  equidistant. 


Given  any  two  lines  in  the  same  plane  and  their  common  per- 
pendicular. If  we  go  out  on  these  lines  in  either  direction  from 
the  perpendicular,  they  approach  each  other.  Now  revolve 
one  of  them  about  this  perpendicular  so  that  they  are  no  longer 
in  the  same  plane.  After  a  certain  amount  of  rotation  the  lines 
will  have  an  infinite  number  of  common  perpendiculars  and  be 
equidistant  throughout  their  entire  length. 

Proof.  Let  p  be  the  length  of  the  common  perpendicular 
AC,  and  take  points  B  and  D  on  the  two  lines  on  the  same 
side  of  this  perpendicular  at  a  distance,  a. 

BD<p,  but  if  CD  revolve  about  AC,  BD  will  become  longer 
than  p  by  the  time  CD  is  revolved  through  a  right  angle ;  for 
BCD  will  then  be  a  right  triangle,  with  BD  for  hypothenuse 
and  BC,  the  hypothenuse  of  the  triangle  ABC,  for  one  of 
its  sides,  so  that  we  shall  have  BD>BC  and  BO  AC. 


PARALLEL   LINES  67 

Suppose,  when  CD  has  revolved  through  an  angle,  6,  BD 
becomes  equal  to^?  and  takes  the  position  BD'.  The  triangles 
ABC  and  D'BC  are  equal,  having  corresponding  sides  equal. 
Therefore,  BD'  is  perpendicular  to  CD'.  BD'  is  also  perpen- 
dicular to  BA  ;  for  if  we  take  the  diedral  A-BC-D'  and  place 
it  upon  itself  so  that  the  positions  of  B  and  C  shall  be  inter- 
changed, A  will  fall  on  the  position  of  D',  and  D'  on  the 
position  of  .4,  and  the  angle  D'BA  must  equal  the  angle  ACD'. 
Therefore,  Hl>'  as  well  as  CA  is  a  common  perpendicular  to 
the  lines  AB  and  CD'. 

Now  at  the  point  C  we  have  a  triedral  whose  three  edges  are 
CB,  CD,  and  CD'.  Moreover,  the  diedral  along  the  edge  CD 
is  a  right  diedral  ;  therefore,  the  three  face  angles  of  the 
triedral  satisfy  the  same  relations  as  do  the  three  sides  of  a 
spherical  right  triangle  ;  namely, 

cos  BCD'  =  cos  BCD  cos  DC  It'. 
But  /;r  /)  =  £-  ACR     and     BCD'  =  ABC. 

Hence,  this  relation  may  be  written 

c-oaABC  =  sin  A  <  'B  cos  B. 
Again,  in  the  right  triangle  ABC 

cos  ABC 


.  '  .  COS  B  =  COS  y>, 
7T 

or,  since  B  and  p  are  less  than  —  ? 

9=  p. 

The  angle  0,  therefore,  does  not  depend  upon  a.  If  we  take 
any  two  lines  in  a  plane  and  turn  one  about  their  common 
perpendicular  through  an  angle  equal  in  measure  to  the  length 


68  NON-EUCLIDEAN   GEOMETRY 

of  that  perpendicular,  the  two  lines  will  then  be  everywhere 
equidistant. 

As  we  have  no  parallel  lines  in  the  ordinary  sense  in  this 
Geometry,  the  name  parallel  has  been  applied  to  lines  of  this 
kind.  They  have  many  properties  of  the  parallel  lines  of 
Euclidean  Geometry. 

Through  any  point  two  lines  can  be  drawn  parallel  to  a 
given  line.  These  are  of  two  kinds,  sometimes  distinguished 
as  right-wound  and  left-wound.  They  lie  entirely  on  a  surface 
of  double  revolution,  having  the  given  line  as  axis.  The  sur- 
face is,  therefore,  a  ruled  surface  and  has  on  it  two  sets  of 
rectilinear  generators  like  the  hyperboloid  of  one  sheet. 


CHAPTER    IV 
ANALYTIC   NON-EUCLIDEAN   GEOMETRY 

WE  shall  use  the  ordinary  polar  coordinates,  p  and  0,  and  for 
the  rectangular  coordinates,  x  and  y,  of  a  point,  we  shall  use 
the  intercepts  on  the  axes  made  by  perpendiculars  through  the 
point  to  the  axes.  The  formulae  depend  upon  the  trigonomet- 
rical relations,  and  in  our  two  Geometries  differ  only  in  the 
use  of  the  imaginary  factor  i  with  lengths  of  lines. 


I.  HYPERBOLIC  ANALYTIC  GEOMETRY 


1.   The  relations  between  polar  and  rectangular  coordinates : 
The  angles  at  the  origin  which  the  radius  vector  makes  with 
the  axes  are  complementary.     From  the  two  right  triangles 

we  have 

tan  ix  =  cos  0  tan  ip, 

tan  iy  =  sin  6  tan  ip. 
Therefore,  tan2  ip  =  tan2  ix  +  tan2  iy, 

tan  iy 

tan  ix 

69 


tan  6    = 


70 


NON-EUCLIDEAN   GEOMETRY 


xy 


2.   The  distance,  8,  between  two  points  : 

cos  i&  =  cos  ip  cos  ip'  +  sin  ip  sin  ip'  cos  (0'  —  6). 

8  and  one  of  the  points  being  fixed,  this  may  be  regarded  as 
the  polar  equation  of  a  circle. 


\ 


3,   The  equation  of  a  line  : 

Let  p  be  the  length  of  the  perpendicular  from  the  origin 
upon  the  line,  and  a  the  angle  which  the  perpendicular  makes 


HYPERBOLIC   ANALYTIC    GEOMETRY 


71 


with  the  axis  of  x.     From  the  right  triangle  formed  with  this 
perpendicular  and  p  we  have 

tan  ip  cos  (0  —  a)  =  tan  ip. 

This  is  the  polar  equation  of  the  line.     We  get  the  equation 
in  x  and  y  by  expanding  and  substituting ;   namely, 

cos  a  tan  ix  +  sin  a  tan  iy  =  tan  ip. 
The  equation    a  tan  ix  +  I  tan  iy  =  i 
represents  a  line  for  which 


tanai> 

Now,  for  real  values  of  p,  —  tan2  ip  <  1  (see  footnote,  p.  59). 
The  line  is  therefore  real  if  a  and  b  are  real,  and  if 


\ 


\ 


4.    The  distance,  8,  of  a  point  from  a  line : 

Let  the  radius  vector  to  the  point  intersect  the  line  at  A, 
and  let  pl  be  the  radius  vector  to  A.      We  have  two  right 


72 


NON-EUCLIDEAN   GEOMETRY 


triangles  with  equal  angles  at  A,  and  from  the  expressions 
for  the  sines  of  these  angles  we  get  the  equation 


sn 


sin  i(p  —  pi) 

This  equation  holds  for  all  points,  xy,  of  the  plane,  8  being 
negative  when  the  point  is  on  the  same  side  of  the  line  as  the 
origin,  and  zero  when  the  point  is  on  the  line. 


Now, 


.  sintp    .    .          .     . 

sin  i&  = r-  sin  ip  —  sin  ip  cos  ip. 

tanip 

= //I  ' 

cos  (0  —  a) 

sin  ip  =  sin  ip  cos  ip  cos  (0  —  a), 


tan  ip1 
and       sin  i&  =  cos  ip  cos  ip  [tan  ip  cos  (0  —  a)  —  tan  ip~\. 

$  being  fixed,  this  may  be  regarded  as  the  polar  equation  of 
an  equidistant-curve. 


\ 


\ 


5.  The  angle  between  two  lines  : 

<f>  being  the  angle  which  a  line  makes  with  the  radius  vector 
at  any  point,  we  have 


THE   ANGLE   BETWEEN   TWO   LINES  73 

cos  <£  =  cos  ip  sin  (0  —  a), 
.        _  sin  ip 
sint'p 

For  two  lines  intersecting  at  this  point, 

sin  i»i  sin  ip» 


sm       sin  <>,  = 


Bin*  ip 

sin «/?!  sin 


=  sm  *P!  sm  ip2  -t 

Now,  from  the  equation  of  the  line 

sin  ipl 

*—  =  cos  i»i  cos  (0  —  «i), 

tantp 

sin  ipa 

±—  =  cos  ipz  cos  (0  —  or2) ; 

tan  ip 

so  that         sin  fa  sin  <£2  =  sin  ipl  sin  ///., 

+  cos  ipi  cos  tp2  cos  (0  —  a  i)  cos  (0  —  or2). 

Again,          cos  ^  cos  fa  =  cos  i^  cos  ip9  sin  (0  —  <*i)  sin  (0  —  or2). 
Adding  these  equations,  we  have 

cos  (<£2  —  <£i)  ==  sin  '(PI  sin  tpa  +  cos  tpi  cos  tp2cos  (o-2  —  a^). 
Two  lines  are  perpendicular  if 

cos  (az  —  a^  +  tan  ipl  tan  ip2  =  0. 
The  lines  a  tan  to;  +  ^»  tan  iy  =  t, 

a'  tan  ix  H-  6'  tan  iy  =  i 
are  perpendicular  if  aa'  +  bb'  =  1. 

6.    The  equation  of  a  circle  in  x  and  y : 

sin  ip  cos  0  =  cos  ip  tan  taj, 
sin  tp  sin  0  =  cos  ip  tan  iy ; 
1  1 


also,      cos  ip  = 


4-  tan2 ip       VI  +  tan2  ix  +  tan2 i?/ 


74 


NON-EUCLIDEAN   GEOMETRY 


The  equation  of  a  circle  may,  therefore,  be  written 
(1  4-  tan2  ix  +  tan2  iy)  (1  4-  tan2  ix'  4-  tan2  iy')  cos2  i8 

=  (1  4-  tan  ix  tan  tic'  +  tan  iy  tan  ty')2. 


7.    The  equation  of  a  boundary-curve  : 

Let  the  axis  of  the  boundary-curve  which  passes  through 
the  origin  make  an  angle,  a,  with  the  axis  of  x9  and  let  the 
point  where  the  boundary -curve  cuts  this  axis  be  at  a  distance, 
k,  from  the  origin,  positive  if  the  origin  is  on  the  convex  side 
of  the  curve,  negative  if  the  origin  is  on  the  concave  side  of 
the  curve.  The  boundary -curve  is  the  limiting  position  of  a 
circle  whose  centre,  on  this  axis,  moves  off  indefinitely. 

p'  being  the  radius  vector  to  the  centre,  the  radius  of  the 
circle  is  p'  —  k,  and  its  equation  may  be  written 

cos  i(p'  —  k)  =  cos  ip  cos  ip'  +  sin  ip  sin  ip'  cos  (6  —  «), 
or,  expanding  and  dividing  by  cos  ip'} 

cos  ik  +  tan  ip'  sin  ik  =  cos  ip  4-  sin  ip  tan  ip1  cos  (0  —  a). 


THE  EQUATION  OF  A  BOUNDARY-CURVE 


75 


Now,  let  p'  increase  indefinitely,     tan  ip1  tends  to  the  limit  i, 
so  that  the  limit  of  the  first  member  of  the  equation  is 

cos  ik  +  i  sin  ik,  or  e~k, 
and  the  polar  equation  of  the  curve  is 

e~k  =  cos  ip  [1  4-  •/  tan  ip  cos  (0  —  nr)] ; 
or,  in  x  y  coordinates, 

(1  +  tan2  ix  4-  tan2  iy)  e~zk 

=  (1  4  i  cos  a  tan  i.r  4  i  sin  a  tan  ty)2. 


Let  &  be  negative  and  equal,  say,  to  —  b,  and  let  a  =  0  ; 
also,  let  a  be  the  ordinate  of  the  point  A  where  the  curve 
cuts  the  axis  of  y. 

Substituting  in  the  equation,  we  find 


=  cos  a. 


Through  A  draw  a  line  parallel  to  the  axis  of  x,  and,  there- 
fore, making  an  angle,  II  (a),  with  the  axis  of  y.  If  we  draw 
a  boundary-curve  through  the  origin  having  the  same  set  of 
parallel  lines  for  axes,  so  that  the  two  boundary-curves  cut 


76  NON-EUCLIDEAN   GEOMETRY 

off  a  distance,  b,  on  these  axes,  we  know  that  the  ratio  of 
corresponding  arcs  is 

s'  1 

-  =  sinll(a)  = ->  (See  p.  56.) 

s  v  '      cosia 


therefore,  -  =  e~b.  (See  p.  45.) 

8.  The  equation  of  an  equidistant-curve  : 

The  polar  equation  of  (4)  reduced  to  an  equation  in  x  and  y 
takes  the  form 

(1  +  tan2  ix  +  tan2  iy)  sin2  iS 

=  cos2  ip  (cos  a  tan  ix  -\-  sin  a  tan  iy  —  tan  ip)2. 

9.  Comparison  of  the  three  equations  : 
The  equation 

(1  +  tan2  ix  +  tan2  iy)  c2  =  —  (i  —  a  tan  ix  —  b  tan  iyf 

represents  a  circle,  a  boundary-curve,  or  an  equidistant-curve, 
according  as  a2  +  b2  <  1,  =1,   >  1,  respectively. 

c 

la 

C 

10.  Differential  formulae : 

Suppose  we  have  an  isosceles  triangle  in  which  the  angle  A 
at  the  vertex  diminishes  indefinitely.     In  the  formula 

sin  A  __  sinC 
sin  ia      sin  ic 

we  may  put  for         sin  A,     sin  ia,     sin  C  ; 

A,  ia,  1, 

respectively.     Therefore, 
(I.) 


DIFFERENTIAL   FORMULA  77 

Corollary.    In  a  circle  of  radius  r,  the  ratio  of  any  arc  to  the 
angle  subtended  at  the  centre  is  sin  ir. 


Again,  in  the  right  triangle  ABC,  let  the  hypothenuse  c 
revolve  about  the  vertex  A.  Differentiating  the  equation 

cos  B 

sm.4  = -> 

cos  tl> 

where  b  is  constant,  we  have 

siuBdB 

cos  Ad  A  = —  - 

COS  ll> 

cos^l 

But  sm  B  = —  > 

cosia 

. ' .  dB  =  —  cos  ia  cos  ib  dA , 
or  (II.)  dB  =  —  cos ic dA. 

Now,  using  polar  coordinates,  we  have  an  infinitesimal  right 
triangle  whose  hypothenuse,  ds,  makes  an  angle,  say  <£,  with 
the  radius  vector  (see  figure  on  page  78).  The  two  sides  about 

the  right  angle  are  dp  and  — r-^  dO ; 

therefore,  ds*  =  dp*  —  sin'2  ip  dO*, 

sin  ip  dO 

tan</>  =  — T-^-T-' 
•t      dp 


78 


NON-EUCLIDEAN   GEOMETRY 


For  two  arcs  cutting  at  right  angles,  let  d'  denote  differ- 
entiation along  the  second  arc : 

sin  ip  dO  _  I      d^p_ 

i      dp 


sinip  d'O 


or 


dp  d'p 

== 


sin2  ip. 


11.    Area: 
It  equals 

We  will  consider  only  the  case  where  the  origin  is  within 
the  area  to  be  computed  and  where  each  radius  vector  meets 
the  bounding  curve  once,  and  only  once. 

Integrating  with  respect  to  p,  from  p  =  0,  we  have 


or 


X2JT 
(cos  ip  —  1)  d&, 
. 

/»27T 

I      cos  ip  dO   -  2  TT. 


*  The  unit  of  area  being  so  chosen  that  the  area  of  an  infinitesimal 
rectangle  may  be  expressed  as  the  product  of  its  base  and  altitude. 


AREA 


79 


Suppose  P  and  P'  are  two   "  consecutive "   points  on  the 
curve,  PM  and  P'M'  the  tangents  at  these  points,  and  <£  the 


angle  which  the  tangent  makes  with  the  radius  vector.     The 
angle  MP'M'  indicates  the  amount  of  turning  or  rotation  at 
these  points  as  we  go  around  the  curve. 
Now,  by  (IT.), 

M/>'M'  =  </<£  +  cos  ipdO. 

In  going  around  the  curve,  <£  may  vary  but  finally  returns 
to  its  original  value.     That  is,  for  our  curve 


=  0, 


COB  <pdO. 


and  the  amount  of  rotation  is 


s: 


Hence,  the  area  is  equal  to  the  excess  over  four  right  angles 
in  the  amount  of  rotation  as  we  go  around  the  curve.  This 
theorem  can  be  extended  to  any  finite  area. 

12.    A  modified  system  of  coordinates  : 

Our  equations  take  simple  forms  if  we  write  lit  for  tan  ix, 
w  for  tan  ///,  lr  for  tan  ip,  and  so  on  for  all  lengths  of  lines. 


80 


NON-EUCLIDEAN   GEOMETRY 


Thus,  we  have  u2  4-  v2  =  r2.* 

The  equation  of  a  line  is 

au  +  bv  =  1, 
and  the  equation 

(1  -  u2  -  v2)  c2  =  (l-  au  -  bv)2 

represents  a  circle,  a  boundary-curve,  or  an  equidistant-curve, 
according  as  a2  +  b2  <  1,   =  1,   >  1,  respectively. 


II.    ELLIPTIC   ANALYTIC   GEOMETRY 

The  Elliptic  Analytic  Geometry  may  be  developed  just  as 
we  have  developed  the  Hyperbolic  Analytic  Geometry,  and  the 
formulae  are  the  same  with  the  omission  of  the  factor  i.  But 
these  formulae  are  also  very  easily  obtained  from  the  relation 
of  line  and  pole,  and  we  shall  produce  them  in  this  way. 

The  formulae  of  Elliptic  Plane  Analytic  Geometry  may  be 
applied  to  a  sphere  in  any  of  our  three  Geometries. 


1.   The  relations  between  polar  and  rectangular  coordinates  : 

tan  x  =  cos  0  tan  p,          tan  y  =  sin  6  tan  p  j 


*  If  we  draw  a  quadrilateral  with  three 
right  angles  and  the  diagonal  to  the  acute 
angle,  and  use  a,  6,  and  c  in  the  same  way  that 
w,  v,  and  r  are  used  above,  the  five  parts 
lettered  in  the  figure  have  the  relations  of  a 


right  triangle  in  the  Euclidean  Geometry  ;  e.g., 


=     ,    etc. 


ELLIPTIC   ANALYTIC   GEOMETRY 

therefore,         tan2  p  =  tan2  x  4  tan'2  y, 


81 


xy 


2.   The  distance,  8,  between  two  points  : 

cos  8  =  cos  p  cos  p'  4  sin  p  sin  p1  cos  (0'  —  0). 
This  may  be  regarded  as  the  polar  equation  of  a  circle  of 
radius  8,  p'  and  6'  being  the  polar  coordinates  of  the  centre. 
Now,  sin  p  cos  0  =  cos  p  tan  #, 

sin  p  sin  0  =  cos  p  tan  y ; 
1  1 


also, 


COSp  = 


1  +  tan2  p       VT+  tan2  x  4  tan2  y 
The  equation  of  a  circle  in  rectangular  coordinates  may,  there- 
fore, be  written 

(1  +  tan2  x  4  tan2  y)  (1  +  tan2  x'  +  tan2  y')  cos2  8 

=  (14-  tan  x  tan  sc'  4-  tan  y  tan  y')2. 

*  The  line  which  has  the  origin  for  pole  forms  with  the  coordinate  axes 
a  trirectangular  triangle,  and  z,  y,  and  6  may  be  regarded  as  representing 
the  directions  of  the  given  point  from  its  three  vertices. 

On  a  sphere,  if  we  take  as  origin  the  pole  of  the  equator,  p  and  0  are 
colatitude  and  longitude,  x  and  y,  one  with  its  sign  changed,  are  the 
"  bearings  "  of  the  point  from  two  points  90°  apart  on  the  equator. 


82 


NON-EUCLIDEAN   GEOMETRY 


3,   The  equation  of  a  line : 

When  8  =  —  >  the  circle  becomes  a  straight  line.     For  this 
we  have,  therefore,  the  equation 

tan  x  tan  x'  +  tan  y  tan  ij  +  1  =  0. 
x'y'  is  the  pole  of  the  line. 
From  the  equation 

tan  p  cos  (6  —  a)  =  tan_/?, 
or  cos  a  tan  x  +  sin  a  tan  y  = 


we  find 


cos  a 

tan  x'  =  — 9 


tan  y  =  — 


sn 


as  can  be  shown  geometrically,  the  polar  coordinates  of  this 

point  being  TT 

P  T  TT»    oc. 

The  equation      a  tan  x  +  b  tan  y  +  1  =  0 
represents  a  real  line  for  any  real  values  of  a  and  b. 

4.    The  distance,  8,  of  a  point  from  a  straight  line  : 

This  is  the  complement  of  the  distance  between  the  point 
and  the  pole  of  the*1  line ;   it  is  expressed  by  the  equation 


DIFFERENTIAL   FOBMULJE 


83 


sin  8  =  —  cos  p  sinp  +  sin  p  eosp  cos  (0  —  a) 
=  cos  p  cos  jt?  [tan  p  cos  (0  —  a)  —  tan  7;]. 

5.   The  angle,  <}>,  between  two  lines  : 

This  is  equal  to  the  distance  between  their  poles  ;  therefore, 

cos  <j>  =  sin  j9  sinp'  4-  cos  p  cos  p1  cos  (a1  —  «). 
The  two  lines     a  tan  x  +  I  tan  y  -f  1  =  0, 
a' tan  a;  +  ft'tany  +  1  =  0 
are  perpendicular  if  aa'  -f  bb1  4-  1  =  0. 


6,   Differential  formulae : 
The  formula 


sn 


sn 


sin  a       sin  c 

becomes,  when  A  diminishes  indefinitely, 
(I.)  a  =  sine-  A. 

Corollary.    In  a  circle  of  radius  r,  the  ratio  of  any  arc  to  the 
*> (bt  ended  at  the  centre  is  sin  r. 


From  the  right  triangle  ABC,  if  If  remain  fixed,  we  get,  by 
differentiating  the  equation 

cos  B 


sin  ^4  = 


cosb 


84  NON-EUCLIDEAX   GEOMETRY 

(II.)  dB  =  -  cose  dA. 


Thus,  we  have  for  differential  formulae  in  polar  coordinates 


tan  <£  =  sin  p  —  >  * 
dp 


*  If  <f>  is  constant,  as  in  the  logarithmic  spiral  of  Euclidean  Geometry, 
we  can  integrate  this  equation  ;  namely, 

tan  0  —  —  =  dd. 
sinp 

.-.  tan  0  log  tan  -  =  0  +  c, 


Writing  <?'  for  etan<^  this  is  tan  -  =  c'  et&n<t>. 

2 

On  the  sphere  this  is  the  curve  called  the  loxodrome. 


A   MODIFIED    SYSTEM   OF   COORDINATES  85 

and  for  two  arcs  cutting  at  right  angles 
dp  d'p 

9*im-*** 

The  formula  for  area  is  * 

J    J  sin  p  dp  <16. 

We  integrate  first  with  respect  to  p,  and  if  the  area  contains 
the  origin  and  each  radius  vector  meets  the  curve  once,  and 
only  once,  our  expression  becomes 


/»27T 

2-7T-  I      cospde. 

*/0 


The  entire  rotation  in  going  around  the  curve  is  found  as 
on  page  79,  and  is 

X2ff 
cos  p  dO. 
_ 

Thus  the  area  is  equal  to  the  amount  by  which  this  rotation 
is  less  than  four  right  angles. 

For  example,  the  area  of  a  circle  of  radius  p  is  2?r  (1  —  cosp), 
and  the  amount  of  turning  in  going  around  it  is  2?r  cos  p.  The 
area  of  the  entire  plane  is  2  TT. 

7.    A  modified  system  of  coordinates  : 

Writing  u  for  tan  x,  v  for  tan  //,  r  for  tan  p,  etc.,  we  have 

u*  +  v2  =  r2.f 

The  equation  of  a  line  then  becomes 
au  +  b»  +  l=  0, 
and  the  equation  of  a  circle 

(1  +  w2  +  ?;2)  c2  =  (1  +  au  +  bv)2. 

*  The  unit  of  area  being  properly  chosen. 
t  The  footnote  on  page  80  applies  here  also. 


86 


NON-EUCLIDEAN   GEOMETRY 


III.    ELLIPTIC    SOLID   ANALYTIC   GEOMETRY 

We  will  develop  far  enough  to  get  the  equation  of  the 
surface  of  double  revolution. 


/ 

T 

A 

A 

O            x 

/ 

1,    Coordinates,  lines,  and  planes  : 

Draw  three  planes  through  the  point  perpendicular  to  the 
axes.  For  coordinates  x,  y,  z,  we  take  the  intercepts  which 
these  planes  make  on  the  axes. 

The  lines  of  intersection  of  these  three  planes  are  perpen- 
dicular to  the  coordinate  planes  (Chap.  I,  II,  16  and  17)  ;  in 
fact,  all  the  face  angles  in  the  figure  are  right  angles  except 
those  at  P  and  the  three  angles  BA'C,  CB'A,  and  AC'B,  which 
are  obtuse  angles. 

Let  p  be  the  radius  vector  to  the  point  P,  and  a,  ft,  and  y 
the  three  angles  which  it  makes  with  the  three  axes.  Then 

cos2  a  -h  cos2  ft  +  cos2  y  =  1, 


cos  a  = 


tana? 


tan/o 
tan-  x  -f  tan2  y  -f  tan2  z  =  tan2  p. 


etc.  ; 


ELLIPTIC   SOLID   ANALYTIC    GEOMETRY  87 

For  the  angle  between  two  lines  intersecting  at  the  origin 
cos  6  =  cos  a  cos  a'  -f  cos  ft  cos  ft'  4-  cos  y  cos  y'. 

The  angle  subtended  at  the  origin  by  the  two  points  xyz 
and  x'y'z'  is  given  by  the  equation 

tan  x  tan  x1  -f-  tan  ?/  tan  ?/'  -f-  tan  z  tan  «' 
cos0  =  -  -*  -  — 

tanp  tan/a 

For  the  distance  between  two  points 

cos  8  =  cos  p  cos  p'  4-  sin  p  sin  p'  cos  0. 

7T 

This  gives  us  the  equation  of  a  sphere,  and  for  8  =  —  the 

A 

equation  of  a  plane.     The  latter  in  rectangular  coordinates  is 
tan  x  tan  x'  +  tan  y  tan  //  4-  tan  z  tan  2'  -f  1  =  0. 

Let  ^>  be  the  length  of  the  perpendicular  from  the  origin 
upon  the  plane,  and  a,  ft,  y  the  angles  which  this  perpendicular 
makes  with  the  axes.  Then  we  have  for  its  pole 


COS  a 

tana-'  =  tan  p'  cos  a  =—       —  >  etc.  : 
tan  jo 

hence,  the  equation  of  the  plane  may  be  written 

cos  a  tan  x  4-  cos  ft  tan  //  4-  cos  y  tan  z  —  tan  p. 

2.   The  surface  of  double  revolution  : 

Take  one  of  its  axes  for  the  axis  of  2,  suppose  k  the  distance 
of  the  surface  from  this  axis,  and  let  6  denote  the  angle  which 
the  plane  through  the  point  P  and  the  axis  of  z  makes  with 
the  plane  of  xz.  We  may  call  z  and  0  latitude  and  longitude. 

Produce  OA  and  CB.    They  will  meet  at  a  distance,  —  >  from 

Ju 

the  axis  of  z  in  a  point,  0',  on  the  other  axis  of  the  surface,  and 
there  form  an  angle  that  is  equal  in  measure  to  z. 


88 


NON-EUCLIDEAN  GEOMETRY 


From  the  right  triangle  O'AB 


But 
and 

Therefore, 
or 

Similarly, 


tan  O'A 

cos  z  = 

tan  O'B 

tan  O'A  =  cot  x, 
tan  O'B  =  cot  CB  = 


cotk 


cosz  = 


tana?  — 


tan  y  — 


cos  6 
tan  k  cos  6 

tana? 
tan  k  cos  6 

COS£ 

tan  k  sin  0 

cosz 


Squaring  and  adding,  we  have  for  the  equation  of  the  surface 
tan2  x  +  tan2  y  =  tan2  k  sec2 «. 


J3 


For  the  length  of  the  chord  joining  two  points  on  the  sur- 
face, we  have 

cos  8  =  cos  p  cos  p'  (1  -f-  tan  x  tan  x'  +  tan  y  tan  y1  -f-  tan  2  tan  «'). 
Now,  tan2  p  =  tan2  &  sec2 «  +  tan2  z ; 

therefore,  sec2  p  —  sec2  &  sec2  *, 

or  cos  p   =  cos  k  cos  *. 


THE  SURFACE  OF  DOUBLE  REVOLUTION      89 

That  is,  in  terms  of  2,  z',  6,  and  0',  we  have 

cos  8  =  cos2  k  cos  («'  —  «)+  sin2  k  cos  (0'  —  0). 
From  this  we  can  get  an  expression  for  ds,  the  differential 
element  of  length  on  the  surface : 

cos  ds  =  cos2  k  cos  dz  +  sin2  k  cos  dO, 

ds2 
or,  since  cos  ds  —  1 —•>  etc., 

a 

ds*  =  cos2  k  dz*  +  sin2  k  dO*. 

z  and  6  are  proportional  to  the  distances  measured  along 
the  two  systems  of  circles.  These  circles  cut  at  right  angles, 
and  may  be  used  to  give  us  a  system  of  rectangular  coordinates 
on  the  surface.  The  actual  lengths  along  these  two  systems  of 
circles  are  Osink  and  2  cos  A:  (see  Cor.  p.  83).  If,  therefore, 
we  write 

a  =  0  sin  k,         ft  =  z  cos  k, 

we  shall  have  a  rectangular  system  on  the  surface  where  the 
coordinates  are  the  distances  measured  along  these  two  systems 
of  circles  which  cut  at  right  angles. 
The  formula  now  becomes 

ds2  =  da2  +  dp. 

An  equation  of  the  first  degree  in  a  and  ft  represents  a  curve 
which  enjoys  on  this  surface  all  the  properties  of  the  straight 
line  in  the  plane  of  the  Euclidean  Geometry.  Through  any 
two  points  one,  and  only  one,  such  line  can  be  drawn,  because 
two  sets  of  coordinates  are  just  sufficient  to  determine  the 
coefficients  of  an  equation  of  the  first  degree.  The  shortest 
distance  between  two  points  on  the  surface  is  measured  on 
such  a  line.  For,  the  distance  between  two  points  on  a  path 
represented  by  an  equation  in  a  and  ft  is  the  same  as  the  dis- 
tance between  the  corresponding  points  and  on  the  correspond- 
ing path  in  a  Euclidean  plane  in  which  we  take  a  and  ft  for 
rectangular  coordinates.  It  must,  therefore,  be  the  shortest 


90  NON-EUCLIDEAN  GEOMETRY 

when  the  path  is  represented  by  an  equation  of  the  first  degree 
in  a  and  ft.  Such  a  line  on  a  surface  is  called  a  geodesic  line, 
or,  so  far  as  the  surface  is  concerned,  a  straight  line.  The 
distance  between  any  two  points  measured  on  one  of  these 
lines  is  expressed  by  the  formula 


Triangles  formed  of  these  lines  have  all  the  properties  of 
plane  triangles  in  the  Euclidean  Geometry :  the  sum  of  the 
angles  is  TT,  etc.  In  fact  this  surface  has  the  same  relation 
to  elliptic  space  that  the  boundary-surface  has  to  hyperbolic 
space. 

The  normal  form  of  the  equation  of  a  line  is 

a.  cos  to  -f-  ft  sin  to  =  p. 

The  rectilinear  generators  of  the  surface  make  a  constant 
angle,  ±  k,  with  all  the  circles  drawn  around  the  axis  which 
is  polar  to  the  axis  of  z.  These  generators  are  then  "straight, 
lines  "  on  the  surface,  and  their  equation  takes  the  form 

a  cos  k  ±  ft  sin  k  =  p. 


HISTORICAL    NOTE 

THE  history  of  Non-Euclidean  Geometry  has  been  so  well 
and  so  often  written  that  we  will  give  only  a  brief  outline. 

There  is  one  axiom  of  Euclid  that  is  somewhat  complicated 
in  its  expression  and  does  not  seem  to  be,  like  the  rest,  a 
simple  elementary  fact.  It  is  this  :  * 

If  two  lines  are  cut  by  a  third,  and  the  sum  of  the  interior 
angles  on  the  same  side  of  the  cutting  line  is  less  than  two 
right  angles,  the  lines  will  meet  on  that  side  when  sufficiently 
produced. 

Attempts  were  made  by  many  mathematicians,  notably  by 
Legendre,  to  give  a  proof  of  this  proposition  ;  that  is,  to  show 
that  it  is  a  necessary  consequence  of  the  simpler  axioms  pre- 
ceding it.  Legendre  proved  that  the  sum  of  the  angles  of  a 
triangle  can  never  exceed  two  right  angles,  and  that  if  there 
is  a  single  triangle  in  which  this  sum  is  equal  to  two  right 
angles,  the  same  is  true  of  all  triangles.  This  was,  of  course, 
on  the  supposition  that  a  line  is  of  infinite  length.  He  could 
not,  however,  prove  that  there  exists  a  triangle  the  sum  of 
whose  angles  is  two  right  angles,  t 

At  last  some  mathematicians  began  to  believe  that  this  state- 
ment was  not  capable  of  proof,  that  an  equally  consistent 

*  See  article  on  the  axioms  of  F.uclid  by  Paul  Tannery,  Bulletin  des 
Sciences  MatMmatiques,  1884. 

t  See,  for  example,  the  twelfth  edition  of  his  Elements  de  Geometrie, 
Livre  I,  Proposition  XIX,  and  Note  II.  See  also  a  statement  by  Klein  in 
an  article  on  the  Non-Euclidean  Geometry  in  the  second  volume  of  the 
first  series  of  the  Bulletin  des  Sciences  Mathematiques. 

91 


92  NON-EUCLIDEAN   GEOMETRY 

Geometry  could  be  built  up  if  we  suppose  it  not  always  true, 
and,  finally,  that  all  of  the  postulates  of  Euclid  were  only 
hypotheses  which  our  experience  had  led  us  to  accept  as 
true,  but  which  could  be  replaced  by  contrary  statements  in 
the  development  of  a  logical  Geometry. 

The  beginnings  of  this  theory  have  sometimes  been  ascribed 
to  Gauss,  biit  it  is  known  now  that  a  paper  was  written  by 
Lambert,*  in  1766,  in  which  he  maintains  that  the  parallel 
axiom  needs  proof,  and  gives  some  of  the  characteristics  of 
Geometries  in  which  this  axiom  does  not  hold.  Even  as  long 
ago  as  1733  a  book  was  published  by  an  Italian,  Saccheri,  in 
which  he  gives  a  complete  system  of  Non-Euclidean  Geometry, 
and  then  saves  himself  and  his  book  by  asserting  dogmatically 
that  these  other  hypotheses  are  false.  It  is  his  method  of 
treatment  that  has  been  taken  as  the  basis  of  the  first  chapter 
of  this  book.t 

Gauss  was  seeking  to  prove  the  axiom  of  parallels  for  many 
years,  and  he  may  have  discovered  some  of  the  theorems  which 
are  consequences  of  the  denial  of  this  axiom,  but  he  never 
published  anything  on  the  subject. 

Lobachevsky,  in  Russia,  and  Johann  Bolyai,  in  Hungary, 
first  asserted  and  proved  that  the  axiom  of  parallels  is  not 
necessarily  true.  They  were  entirely  independent  of  each 
other  in  their  work,  and  each  is  entitled  to  the  full  credit  of 
this  discovery.  Their  results  were  published  about  1830. 

It  was  a  long  time  before  these  discoveries  attracted  much 
notice.  Meanwhile,  other  lines  of  investigation  were  carried 
on  which  were  afterwards  to  throw  much  light  on  our  subject, 
not,  indeed,  as  explanations,  but  by  their  striking  analogies. 

Thus,  within  a  year  or  two  of  each  other,  in  the  same 
journal  (Crelle)  appeared  an  article  by  Lobachevsky  giving 

*  See  American  Mathematical  Monthly,  July-August,  1895. 
t  The  translation  of  Saccheri  by  Halsted  has  been  appearing  in  the 
American  Mathematical  Monthly. 


HISTORICAL   NOTE  98 

the  results  of  his  investigations,  and  a  memoir  by  Minding  on 
surfaces  on  which  he  found  that  the  formulae  of  Spherical 
Trigonometry  hold  if  we  put  ia  for  a,  etc.  Yet  these  two 
papers  had  been  published  thirty  years  before  their  connection 
was  noticed  (by  Beltrami). 

Again,  Cayley,  in  1859,  in  the  Philosophical  Transactions, 
published  his  Sixth  Memoir  on  Quantics,  in  which  he  developed 
a  projective  theory  of  measurement  and  showed  how  metrical 
properties  can  be  treated  as  projective  by  considering  the 
anharmonic  relations  of  any  figures  with  a  certain  special 
figure  that  he  called  the  absolute.  In  1872  Klein  took  up 
this  theory  and  showed  that  it  gave  a  perfect  image  of  the 
Non-Euclidean  Geometry. 

It  has  also  been  shown  that  we  can  get  our  Non-Euclidean 
Geometries  if  we  think  of  a  unit  of  measure  varying  according 
to  a  certain  law  as  it  moves  about  in  a  plane  or  in  space. 

The  older  workers  in  these  fields  discovered  only  the 
Geometry  in  which  the  hypothesis  of  the  acute  angle  is 
assumed.  It  did  not  occur  to  them  to  investigate  the  assump- 
tion that  a  line  is  of  finite  length.  The  Elliptic  Geometry 
was  left  to  be  discovered  by  Eiemann,  who,  in  1854,  took  up  a 
study  of  the  foundations  of  Geometry.  He  studied  it  from 
a  very  different  point  of  view,  an  abstract  algebraic  point  of 
view,  considering  not  our  space  and  geometrical  figures,  except 
by  way  of  illustration,  but  a  system  of  variables.  He  investi- 
gated the  question,  What  is  the  nature  of  a  function  of  these 
variables  which  can  be  called  element  of  length  or  distance  ? 
and  found  that  in  the  simplest  cases  it  must  be  the  square 
root  of  a  quadratic  function  of  the  differentials  of  the  varia- 
bles whose  coefficients  may  themselves  be  functions  of  the 
variables.  By  taking  different  forms  of  the  quadratic  expres- 
sions we  get  an  infinite  number  of  these  different  kinds  of 
Geometry,  but  in  most  of  them  we  lose  the  axiom  that  bodies 
may  be  moved  about  without  changing  their  size  or  shape. 


94  NON-EUCLIDEAN   GEOMETRY 

Two  more  names  should  be  included  in  this  sketch,  —  Helm- 
holtz  and  Clifford.  These  did  much  to  make  the  subject 
popular  by  articles  in  scientific  journals.  To  Clifford  we  owe 
the  theory  of  parallels  in  elliptic  space,  as  explained  on  page  68. 
He  showed  that  we  can  have  in  this  Geometry  a  finite  surface 
on  which  the  Euclidean  Geometry  holds  true.* 

The  chief  lesson  of  Non-Euclidean  Geometry  is  that  the 
axioms  of  Geometry  are  only  deductions  from  our  experience, 
like  the  theories  of  physical  science.  For  the  mathematician, 
they  are  hypotheses  whose  truth  or  falsity  does  not  concern 
him,  but  only  the  philosopher.  He  may  take  them  in  any  form 
he  pleases  and  on  them  build  his  Geometry,  and  the  Geome- 
tries so  obtained  have  their  applications  in  other  branches  of 
mathematics. 

The  "axiom,"  so  far  as  this  word  is  applied  to  these  geo- 
metrical propositions,  is  not  "  self-evident,7'  and  is  not  neces- 
sarily true.  If  a  certain  statement  can  be  proved,  —  that  is.  if 
it  is  a  necessary  consequence  of  axioms  already  adopted,  —  then 
it  should  not  be  called  an  axiom.  When  two  or  more  mutually 
contradictory  statements  are  equally  consistent  with  all  the 
axioms  that  have  already  been  accepted,  then  we  are  at  liberty 
to  take  either  of  them,  and  the  statement  which  we  choose 

*  Some  of  the  more  interesting  accounts  of  Non-Euclidean  Geometry 
are:  Encyclopedia  Britannica,  article  "Measurement/*  by  Sir  Robert  Ball. 
Recue  Generate  des  Sciences,  1891,  "  Les  Geometries  Non-Euclidean.*'  by 
Poincare\  Bulletin  of  the  American  Mathematical  Society,  May  and  June, 
1900,  "  Lobachevsky's  Geometry,"  by  Frederick  S.  Woods.  Mathema- 
tiscfie  Annalen,  Bd.  xlix,  p.  149,  1897,  and  Bulletin  des  Sciences  Mathe- 
matiques,  1897,  "Letters  of  Gauss  and  Bolyai'*;  particularly  interesting 
is  one  letter  in  which  Gauss  gives  a  formula  for  the  area  of  a  triangle  on 
the  hypothesis  that  we  can  draw  three  mutually  parallel  lines  enclosing  a 
finite  area  always  the  same.  The  last  two  articles  refer  to  the  publica- 
tions of  Professors  Engel  and  Stackel,  which  give  in  German  a  full  history 
of  the  theory  of  parallels  and  the  writings  and  lives  of  Lobachevsky  and 
Bolyai  See  also  the  translations  by  Prof.  George  Bruce  Halsted  of 
Lobachevsky  and  Bolyai  and  of  an  address  by  Professor  Vasiliev. 


HISTORICAL   *OTE  95 

becomes  for  our  Geometry  an  axiom.     Our  Geometry  is  a  study 
of  the  consequences  of  this  axiom. 

The  assumptions  which  distinguish  the  three  kinds  of  Geom- 
etry that  we  have  been  studying  may  be  expressed  in  different 
forms.  We  may  say  that  one  or  two  or  no  parallels  can  be 
drawn  through  a  point ;  or,  that  the  sum  of  the  angles  of  a 
triangle  is  equal  to,  less  than,  or  greater  than  two  right  angles  ; 
or,  that  a  straight  line  has  two  real  points,  one  real  point,  or 
no  real  point  at  infinity ;  or,  that  in  a  plane  we  can  have 
similar  figures  or  we  cannot  have  similar  figures,  and  a  straight 
line  is  of  finite  or  infinite  length,  etc.  But  any  of  these  forms 
determines  the  nature  of  the  Geometry,  and  the  others  are 
deducible  from  it. 


ADVERTISEMENTS 


PLANE    AND   SOLID 

Analytic  Geometry 

By  FREDERICK  H.  BAILEY,  A.M.  (Harvard),  and  FREDERICK 

S.  WOODS,  Ph.D.  (Gottingen),  Assistant  Professors 

of  Mathematics  in  Massachusetts  Institute 

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8vo.   Cloth.   371  pages.   For  introduction,  $2.00. 


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In  the  solid  geometry,  besides  the  plane  and  the  straight 
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Wentworth's  Plane  Geometry  (Revised) 

Wentworth's  Solid  Geometry  (Revised)  

Wentworth's  Plane  and  Solid  Geometry  (Revised)  and  Plane  Trig. 
Wentworth's  Analytic  Geometry. 


.00 

.12 
.12 

40 

•5° 
•25 
•75 
•75 
.40 

•25 

Wentworth  and  Hill's  Five-place  Log.  and  Trig.  Tables  (7  tables)       .50 
Wentworth  and  Hill's  Five-place  Log.  and  Trig.  Tables  (complete 

edition) i-oo 

Wheeler's  Plane  and  Spherical  Trigonometry  and  Tables i.oo 

GINN   &   COMPANY,  Publishers, 

Boston.      New  York.     Chicago.      San  Francisco.     Atlanta.      Dallas. 
Columbus.      London. 


14  DAY  USE 

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176B 


General  Library 

University  of  California 

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